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The Design of Renewable Fuel Mandates and Cost Containment Mechanisms

Abstract

Policies to reduce greenhouse gas emissions from transportation fuels often take the form of renewable fuel mandates rather than taxes or cap-and-trade programs. Delays in the development and deployment of new technologies when binding mandates exist for their use may lead to situations with high compliance costs. We study the effects and efficiency of two mandates, a renewable share mandate and a carbon intensity standard, with and without a cost containment mechanism. Using both a theoretical model of a regulated fuel industry and a numerical model of the U.S. fuel market, we show that cost containment mechanisms can have the benefit of both constraining compliance costs and limiting deadweight loss. According to our numerical results, an optimally set mandate alone leads to only modest gains over business as usual welfare levels. The efficiency of both policies, especially carbon intensity standards, can increase substantially when combined with a cost containment mechanism.

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Fig. 1

Notes

  1. In addition, because the feedstocks used for the production of corn-based ethanol can also be used for food, there is a related literature on the effects of ethanol policies on the relationship between food and fuel markets (Runge and Senauer 2007; Rajagopal et al. 2007; Wright 2014; Poudel et al. 2012; Abbott et al. 2008, 2009, 2011; de Gorter et al. 2013; Filip et al. 2018).

  2. Although the RFS already has a cost containment mechanism similar to the one considered here (EPA 2017), the current mechanism applies only to the mandate’s cellulosic portion and has played only a limited role in containing compliance costs in the program.

  3. While we focus on renewable fuel mandates, cost containment mechanisms are also prevalent in cap-and-trade policies. For example, California amended its cap-and-trade program to include a price ceiling (CARB 2018). The revised Regional Greenhouse Gas Initiative (RGGI) model rule, which took into effect in 2014, added a cost containment reserve (CCR) to the cap-and-trade system (Ramseur 2019). As part of its 2016 program review, RGGI considered a new provision, known as an emissions containment reserve (ECR), that would incorporate a minimum price for specified quantities of allowances under the cap and introduce steps into the allowance supply function (Burtraw et al. 2017). The revised European Union Emissions Trading System (EU ETS) also includes cost containment to mitigate excessive price fluctuations in the allowance market. The provision began in 2013 and applies if, for more than six consecutive months, the allowance price is more than three times the average price of allowances during the two preceding years on the European market (Commission European 2020).

  4. Many renewable portfolio standards already include cost containment mechanisms, such as Alternative Compliance Payments in Massachusetts (Mass.gov 2021), New Jersey (DSIRE 2018), Ohio (PUCO 2020), Maryland, and Delaware, among other states (Bird et al. 2011; Barbose 2019).

  5. In addition to the federal RFS, state-level renewable share mandates have been implemented in Oregon, Washingon, Minnesota, Missouri, Louisiana, and Pennsylvania (Center for Climate and Energy Solutions 2019).

  6. If met, the policy would require around 25% of motor gasoline to contain ethanol.

  7. Cellulosic biofuels are fuels produced from non-edible biomass such as corn stover or switchgrass. Biomass-based diesel is produced mostly from animal fats or vegetable oils such as soybean oil. Biofuels qualify as ‘advanced’ if their lifecycle greenhouse gas emissions are below a threshold set by the EPA.

  8. California’s LFCS is an example of a carbon intensity standard since it requires that the average carbon intensity of fuels be below a specified standard. While California’s LCFS is the largest carbon intensity standard for transportation fuels, British Columbia and Oregon have similar policies in place, and Washington and the European Union have proposed instituting low carbon fuel standards (British Columbia Ministry of Energy and Mines 2014; Oregon Department of Environmental Quality 2016; Pont et al. 2014; Commission European 2014).

  9. Carbon intensity (CI) values represent the ARB’s estimate of the carbon equivalent emissions rate of a given fuel’s life-cycle production process.

  10. RINs are differentiated by vintage year and fuel type to enforce banking restrictions and ensure the mandate for each biofuel category is met.

  11. For example, every gallon of gasoline sold generates a deficit equal to the difference between gasoline’s CI and the standard. Analogously, every gallon of fuel produced that has a lower CI than the standard generates a credit surplus equal to the difference between the standard and its CI. Thus, credits (deficits) are generated only for the amount of emissions below (above) the standard.

  12. Recent years have seen increased attention given to E15, fuel containing up to 15% ethanol. If widely adopted, E15 could ‘break’ the current blend wall since most gasoline-powered vehicles in use today can use the fuel. Nevertheless, for many reasons, its use has been very limited to date (Lade 2019).

  13. Obligated parties purchased just under 12 million credits in 2017, the latest year for which the EPA has reported data of annual credits purchased. In that year, total cellulosic biofuel obligations were nearly 288 million gallons, and total renewable fuel obligations were 17.8 billion gallons. The highest reported amount of cellulosic waiver credits purchased was in 2016 when producers purchased just over 33 million credits (EPA 2020, 2021).

  14. Lade and Lin (2013) compare the effectiveness of each of the ARB’s proposals in constraining compliance costs under the LCFS. A low carbon credit multiplier acts similarly to relaxing the policy constraint, and is therefore not considered here.

  15. The assumption of two inputs is made for notational ease. While the qualitative results are not affected in the multi-fuel case (Lade and Lin 2013), not all the analytic results presented in this section generalize to the multi-fuel case. For example, a numerical equivalence exists between a renewable fuel share mandate and carbon intensity standard in the two-fuel case. The equivalence breaks down in the multi-fuel case. We examine the multi-fuel case in our numerical model in Section 4.

  16. Denominating in gasoline gallon equivalent (GGE) units may be needed if consumers value the energy content of fuel, since a gallon of renewable fuel typically does not contain as much energy as a gallon of gasoline or diesel. A gallon of ethanol has around 70% of the energy content of a gallon of gasoline. A gallon of biomass-based diesel has around 95% of the energy content of conventional diesel.

  17. Given our assumption of a single renewable fuel, we do not model the RFS using a nested mandate structure. Thus, our model is most applicable to the overall biofuel mandate under the RFS.

  18. The proof follows Montgomery (1972): when firms can trade compliance credits, have perfect information, and face no trading costs, marginal compliance costs are equalized to \(\lambda\) across all firms.

  19. We analyze the effects of each fuel mandate on important market outcomes in Appendix A (Section 6). Appendix B.1 (Section 7.1) provides a more rigorous analysis of why fuel mandates cannot achieve the first-best outcome. Appendix B.2 (Section 7.2) present a graphical analysis of the second-best nature of fuel mandates.

  20. We examine the market effects of a cost containment mechanism that caps compliance costs in Appendix A (Section 6). Proposition A.2 in Appendix A (Section 6) summarizes the comparative statics for the credit window price \({\bar{p}}^{\text {cred}}\) when firms purchase from the credit window.

  21. The same would remain true for a nested renewable share mandate structure. While the nested mandate levels and credit window prices may be adjusted, yielding efficiency gains beyond using only an overall mandate, the lowest-tiered mandate will always serve as an implicit subsidy for the lowest tiered renewable fuel, preventing the renewable share mandate policy from achieving the first-best.

  22. Renewable diesel has become an important compliance fuel in California in recent years since renewable diesel is chemically very similar to diesel and therefore faces fewer blending constraints than biodiesel does (EIA 2020). Nevertheless, the carbon intensities for biodiesel and renewable diesel are very similar, and we lack evidence that they have differential supply elasticities (EIA 2018c). Given this, we aggregate biodiesel and renewable diesel in this exercise.

  23. Our focus on longer-term scenarios also motivates our choice not to model the blend wall for either motor gasoline or motor diesel fuel as others have recently done (e.g., Korting et al. (2019)).

  24. The 2019 RFS2 mandates for cellulosic ethanol were 418 mgals. Most RINs produced to satisfy the cellulosic mandate in that year were produced from biogas and other non-liquid cellulosic ethanol producers, however (EIA 2018b). Thus, despite assuming low values for future cellulosic production, our assumptions here are likely quite generous for liquid cellulosic ethanol production.

  25. Recall that, with a credit window, the implicit gasoline tax is \(\alpha \times {\bar{p}}^{\text {cred}}\) and \((\phi ^c-\sigma ) {\bar{p}}^{\text {cred}}\) for the RFS and LCFS, respectively.

  26. Results for the low cellulosic/biodiesel scenario are broadly similar.

  27. Recall that the business as usual average CI is 1 in our model. Thus the second-best LCFS of \(\sigma =0.952\) corresponds to a policy requiring a \((1-0.952) \times 100\) % average CI reduction.

  28. Figure 4 in Appendix C (Section 8) shows credit purchases in each scenario in panel (b). Credit window purchases decrease as credit window prices become higher, and reach zero when credit window prices exceed the policy constraint’s shadow price.

  29. Moreover, for the special case when \(\alpha =-(\sigma -\phi ^c)/(\sigma -\phi ^r)\), the policy functions \(\varphi (q^c, q^r;\theta )\) for both the RFS and the LCFS are identical.

  30. Note that \(\xi ^c\) and \(\eta ^d\) represent local elasticities.

  31. To confirm this, note that \(\text {det}(H) = \frac{1}{\xi ^c}\frac{P}{q^c} - \frac{1}{\eta ^d} \frac{P}{Q} + \alpha ^2 \left( \frac{1}{\xi ^r}\frac{P}{q^r} - \frac{1}{\eta ^d}\frac{P}{Q} \right) - 2 \alpha \frac{1}{\eta ^d} \frac{P}{Q} > 0\) for the RFS and \(\text {det}(H) =(\sigma -\phi ^c)^2 \left( \frac{1}{\xi ^r} \frac{P}{q^r} - \frac{1}{\eta ^d} \frac{P}{Q} \right) + (\sigma -\phi ^r)^2 \left( \frac{1}{\xi ^c}\frac{P}{q^c} - \frac{1}{\eta ^d} \frac{P}{Q} \right) + 2(\sigma -\phi ^c)(\sigma -\phi ^r) \frac{1}{\eta ^d} \frac{P}{Q} >0\) for the LCFS.

  32. Fischer (2010) derives analogous results for the effect of Renewable Portfolio Standards on wholesale electricity prices.

  33. Specifically, the solid circles represent level curves of the function \(U(Q)-C^c(q^c)-C^r(q^r)\), and the dashed circles represent level curves of the function \(U(Q)-C^c(q^c)-C^r(q^r)-D(\phi ^c q^c + \phi ^r q^r)\), where U(Q) is the total benefit from fuel consumption. When the marginal utility of income is constant, the total benefit from fuel consumption Q is given by the area under the demand curve, which measures the gross consumer surplus. Weitzman (2003) shows that using the area under the demand curve in place of firm revenue yields the same outcome as a perfectly competitive market (Lin and Wagner 2007; Lin 2009; Lin et al. 2009; Lin Lawell 2021).

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Acknowledgements

We thank Sonia Yeh and the California Air Resources Board for generously funding our research. This project was also supported by the USDA National Institute of Food and Agriculture, Hatch project number CA-D-ARE-2200-H; and by a grant from the National Center for Sustainable Transportation, which is supported by the U.S. Department of Transportation through the University Transportation Centers program. We received helpful comments from James Bushnell, Meredith Fowlie, Aaron Smith, Jim Wilen, Kevin Novan, Billy Pizer, Derek Nixon, John Courtis, Sonia Yeh, Julie Witcover, Linda Nøstbakken, Debraj Ray, and seminar participants at the UC-Davis Environmental and Resource Economics Workshop, the Stanford University Precourt Energy Efficiency Center Sustainable Transportation Seminar, and the Berkeley Bioeconomy Conference. Lin Lawell is a former member of the Giannini Foundation of Agricultural Economics and a Faculty Fellow at the Atkinson Center for a Sustainable Future. All errors are our own.

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Appendices

Market Effects

Market Effects of Fuel Mandates

We first study the effects of each fuel mandate on important market outcomes. The compliance credit price under both an RFS and an LCFS is driven by the differences in marginal cost between the renewable and conventional fuel. To see this, combine the two optimality conditions (1) and (2) for each mandate to yield:

$$\begin{aligned} \begin{array}{cc} \text {[RFS:]} &{} \lambda = \frac{\frac{\partial C^r}{\partial q^r} - \frac{\partial C^c}{\partial q^c}}{1+\alpha } \\ \text {[LCFS:]} &{} \lambda = \frac{\frac{\partial C^r}{\partial q^r} -\frac{\partial C^c}{\partial q^c}}{\phi ^c-\phi ^r} . \end{array} \end{aligned}$$

The conditions state that \(\lambda\) equals the weighted difference between the renewable and conventional fuel marginal costs. Thus, all else equal a high spread between marginal costs of renewable and conventional fuels will lead to high compliance costs under either policy.

Equilibrium fuel prices under each mandate equal a weighted average of the marginal costs of each fuel, where the weights correspond to the share requirement under each respective mandate. To see this, substitute each solution for \(\lambda\) above into either Eqs. (1) or (2) to obtain:

$$\begin{aligned} \begin{array}{cc} \text {[RFS:]} &{} P=\frac{1}{1+\alpha } \frac{\partial C^c}{\partial q^c} + \frac{\alpha }{1+\alpha }\frac{\partial C^r}{\partial q^r} \\ \\ \text {[LCFS:]} &{} P= \frac{\sigma - \phi ^r}{\phi ^c - \phi ^r}\frac{\partial C^c}{\partial q^c} + \frac{\phi ^c - \sigma }{\phi ^c - \phi ^r} \frac{\partial C^r}{\partial q^r} . \end{array} \end{aligned}$$

The equations illustrate the similarity of the two fuel mandates.Footnote 29 The distinguishing factor between the policies is how the share mandate is constructed. The RFS share mandate is explicitly set by \(\alpha\), while the LCFS share mandate is implicitly determined by the fuels’ relative carbon intensity factors.

Proposition A.1 summarizes important comparative statics with respect to the policy parameters under a binding fuel mandate.

Proposition A.1

Market effects of fuel mandates

  1. 1.

    Under both mandates, increasing the stringency of the policy reduces production of the conventional fuel \(q^c\).

  2. 2.

    Under an RFS, increasing \(\alpha\) increases production of renewable fuel \(q^r\) if \(\frac{1}{\xi ^c} - \frac{1}{\eta ^d} > \alpha \frac{ \lambda }{P}\), where \(\xi ^c\) is the price elasticity of supply for the conventional input and \(\eta ^d\) is the elasticity of demand.Footnote 30 Under an LCFS, decreasing \(\sigma\) increases production of renewable fuel \(q^r\) if \(\frac{1}{\xi ^c} - \frac{1}{\eta ^d} > (\phi ^c - \sigma ) \frac{\lambda }{P}\).

Proof of Proposition A.1

Taking the total differential of equations (1), (2) and the policy constraint yields:

$$\begin{aligned} \underbrace{ \left[ \begin{array}{ccc} \frac{\partial P}{\partial Q} - \frac{\partial ^2 C^c}{\partial q^c \partial q^c} &{}\frac{\partial P}{\partial Q} &{} \frac{\partial \varphi (\cdot )}{\partial q^c} \\ \frac{\partial P}{\partial Q} &{}\frac{\partial P}{\partial Q} - \frac{\partial ^2 C^r}{\partial q^r \partial q^r} &{} \frac{ \partial \varphi (\cdot )}{\partial q^r} \\ \frac{\partial \varphi (\cdot )}{\partial q^c} &{}\frac{ \partial \varphi (\cdot )}{\partial q^r} &{} 0 \end{array} \right] }_{=H} \left[ \begin{array}{c} dq^c \\ dq^r \\ d \lambda \end{array} \right] = \underbrace{ \left[ \begin{array}{c} -\lambda \frac{ \partial ^2 \varphi (\cdot )}{\partial q^c \partial \theta } \\ -\lambda \frac{\partial ^2 \varphi (\cdot )}{\partial q^r \partial \theta } \\ - \frac{\partial \varphi (\cdot )}{\partial \theta } \end{array} \right] }_{=h} d \theta . \end{aligned}$$

Let \(\eta ^d\) denote the price elasticity of demand for fuel and \(\xi ^i\) denote the price elasticity of supply for \(i=c,r\). Substituting \(\frac{\partial P}{\partial Q} = \frac{1}{\eta ^d} \frac{P}{Q}\), and \(\frac{\partial ^2 C^i}{\partial q^i \partial q^i} = \frac{1}{\xi ^i} \frac{P}{q^i}\) for \(i=c,r\):

$$\begin{aligned} \underbrace{ \left[ \begin{array}{ccc} \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^c} \frac{P}{q^c} &{} \frac{1}{\eta ^d} \frac{P}{Q} &{} \frac{\partial \varphi (\cdot )}{\partial q^c} \\ \frac{1}{\eta ^d} \frac{P}{Q} &{} \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} &{} \frac{ \partial \varphi (\cdot )}{\partial q^r} \\ \frac{\partial \varphi (\cdot )}{\partial q^c} &{}\frac{ \partial \varphi (\cdot )}{\partial q^r} &{} 0 \end{array} \right] }_{=H} \left[ \begin{array}{c} dq^c \\ dq^r \\ d \lambda \end{array} \right] = \underbrace{ \left[ \begin{array}{c} -\lambda \frac{ \partial ^2 \varphi (\cdot )}{\partial q^c \partial \theta } \\ -\lambda \frac{\partial ^2 \varphi (\cdot )}{\partial q^r \partial \theta } \\ - \frac{\partial \varphi (\cdot )}{\partial \theta } \end{array} \right] }_{=h} d \theta . \end{aligned}$$

The matrix H is the bordered Hessian and is negative semi-definite by concavity of the objective function. We can solve for \(\frac{dx}{d \theta }\) for \(x \in \{q^c, q^r \}\) using Cramer’s rule:

$$\begin{aligned} \frac{d x}{d \theta } = \frac{ \text {det}(H^i)}{\text {det}(H)} , \end{aligned}$$

where H is the bordered Hessian and \(H^i(\cdot )\) is the matrix H with the ith column replaced with column h. Note that \(\text {det}(H) > 0\) for both policies.Footnote 31 Thus, the signs of the effects are determined by \(\text {sign} \left( \text {det}(H^i) \right)\).

Solving for the RFS yields:

$$\begin{aligned} \frac{ dq^c}{d \alpha }&= \left( \frac{P}{\eta ^d} - \frac{P}{\xi ^r} - \lambda \right) \text {det}(H)^{-1} <0 \\ \frac{ dq^r}{d \alpha }&= \left( \frac{P}{\xi ^c} - \frac{P}{\eta ^d} - \alpha \lambda \right) \text {det}(H)^{-1}. \end{aligned}$$

Considering the price effect:

$$\begin{aligned} \frac{d P}{d \alpha }&=\frac{\partial P}{\partial Q} \left( \frac{dq^c}{d \alpha } + \frac{dq^r}{d \alpha } \right) \\&= \frac{1}{\eta ^d} \frac{P}{Q} \frac{dQ}{d\alpha }. \end{aligned}$$

Solving for the LCFS yields:

$$\begin{aligned} \frac{dq^c}{d \sigma }&=\left( (\phi ^c-\phi ^r) \left( \frac{P}{\xi ^r} - \frac{P}{\eta ^d} \right) + (\sigma -\phi ^r)(\phi ^c-\phi ^r) \lambda \right) \text {det}(H)^{-1} >0 \\ \frac{d q^r}{d \sigma }&= \left( (\phi ^c-\phi ^r) \left( \frac{P}{\eta ^d} - \frac{P}{\xi ^c} \right) + (\phi ^c-\sigma ) (\phi ^c-\phi ^r) \lambda \right) \text {det}(H)^{-1}. \end{aligned}$$

with fuel price effects:

$$\begin{aligned} \frac{d P}{d \sigma } = \frac{1}{\eta ^d}\frac{P}{Q} \frac{d Q}{d\sigma }. \end{aligned}$$

Increasing the stringency of both policies decreases \(q^c\) and increases \(q^r\) so long as \(\frac{1}{\xi ^c}- \frac{1}{\eta ^d}\) is larger than a term proportional to the ratio of the compliance credit price and the fuel price, \(\frac{ \lambda }{P}\). Thus, if the supply of conventional fuel or the demand for fuel are relatively inelastic, increasing the stringency of either policy will increase \(q^r\). Intuitively, if consumers do not decrease consumption or conventional suppliers do not reduce their supply as the policies become more stringent, the only means to maintain compliance is to increase the supply of renewable fuel. If, however, consumers reduce fuel consumption or fossil fuel producers reduce production in response to increases in the stringency of the policies, \(q^r\) does not necessarily need to increase to maintain compliance. The effect of the mandates on fuel prices depends on total fuel supply response. It can be shown that a necessary condition for the policy to increase fuel price P is \(\xi ^c > \xi ^r\), i.e., fuel prices increase as the policies become more stringent if the supply elasticity of the conventional fuel is greater than the renewable supply elasticity.Footnote 32

Figure 2 illustrates the effects of both policies. The left figure graphs the no policy equilibrium, and the right figure graphs equilibrium under a fuel mandate. In both graphs, the downward sloping line is the fuel demand curve; the upward sloping line with triangles is the conventional fuel supply curve; the upward sloping line with circles is the renewable fuel supply curve; and the bold upward sloping line is the total fuel supply curve, equal to the horizontal sum of the renewable and conventional supply curves.

In the left figure, the total and conventional fuel supply curves are the same until the price reaches the intercept of the renewable fuel supply curve. The initial market-clearing price \(P_0\) and total fuel quantity \(Q_0\) are found where the total fuel supply curve intersects the demand curve. The supply of conventional and renewable fuel, \(q_0^c\) and \(q_0^r\), respectively, is given by the corresponding quantity where the equilibrium price intersects the individual supply functions.

Fig. 2
figure 2

Market effects of fuel mandates*. The left figure illustrates the no policy equilibrium, and the right graphs the equilibrium under a fuel mandate. The downward sloping line is the fuel demand curve, the upward sloping line with triangles is the conventional fuel supply curve, the upward sloping line with circles is the renewable fuel supply curve, and the bold upward sloping line is the total fuel supply curve

The right graph illustrates equilibrium under a binding fuel mandate, with the solid lines representing the initial supply curves and the dashed lines representing the supply curves net of the fuel mandate’s implicit subsidy and tax. Under both policies, the renewable supply curve shifts down, and the conventional supply curve shifts up until the market-clearing price and quantities are such that the equilibrium quantities comply with the mandate. The equilibrium price \(P_M\) and quantity \(Q_M\) are found where the new dashed total fuel supply curve, equal to the sum of the shifted conventional and renewable supply curves, intersects the demand curve. In our example, the resulting equilibrium results in greater production of renewable fuel \(q^r_M\) and lower production of conventional fuel \(q^c_M\). Because total fuel consumption \(Q_M\) declines, the policy results in higher fuel prices \(P_M\) over the no policy equilibrium.

Market Effects of Cost Containment Mechanism

We next examine the market effects of a cost containment mechanism that caps compliance costs. We model the cost containment mechanism as the regulator offering a credit window for compliance credits. Under a credit window, firms have the option to purchase compliance credits directly from the regulator at a given credit window price. The credits purchased from the regulator would not be associated with the production of any renewable fuel but would increase liquidity in the market.

Let \(c>0\) denote the number of credits bought from the regulator through the window and \({\bar{p}}^{\text {cred}}\) be the credit window price. Proposition A.2 summarizes the comparative statics for the credit window price \({\bar{p}}^{\text {cred}}\) when firms purchase from the credit window.

Proposition A.2

Suppose firms purchase from the credit window such that \(\lambda ={\bar{p}}^{\text {cred}}\). Under both fuel mandates as the credit window price \({\bar{p}}^{\text {cred}}\) increases:

  1. i

    The volume \(q^c\) of conventional fuel decreases and the volume \(q^r\) of renewable fuel increases; and

  2. ii

    The quantity c of compliance credits decreases.

Proof of Proposition A.2

Taking the total differential of equations (1), (2), and the policy constraint yields:

$$\begin{aligned} \underbrace{ \left[ \begin{array}{ccc} \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^c} \frac{P}{q^c} &{} \frac{1}{\eta ^d} \frac{P}{Q} &{} 0 \\ \frac{1}{\eta ^d} \frac{P}{Q} &{} \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} &{} 0 \\ \frac{\partial \varphi (\cdot )}{\partial q^c} &{}\frac{ \partial \varphi (\cdot )}{\partial q^r} &{} 1 \end{array} \right] }_{=H} \left[ \begin{array}{c} dq^c \\ dq^r \\ d c \end{array} \right] = \underbrace{ \left[ \begin{array}{c} -\frac{ \partial \varphi (\cdot )}{\partial q^c} \\ -\frac{ \partial \varphi (\cdot )}{\partial q^r} \\ 0 \end{array} \right] }_{=h} d {\bar{p}}^{\text {cred}}. \end{aligned}$$

We can derive the comparative statics using Cramer’s rule. For the RFS, this yields:

$$\begin{aligned} \frac{dq^c}{d {\bar{p}}^{\text {cred}}}&= \left( \frac{1}{\eta ^d} \frac{P}{Q} + \alpha \left( \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} \right) \right) \text {det}(H)^{-1} <0 \\ \frac{d q^r}{d {\bar{p}}^{\text {cred}}}&= \left( \frac{1}{\eta ^c} \frac{P}{q^c} - (1+\alpha ) \frac{1}{\eta ^d} \frac{P}{Q} \right) \text {det}(H)^{-1} >0 \end{aligned}$$
$$\begin{aligned} \frac{d c}{ d {\bar{p}}^{\text {cred}} } = \left( \alpha ^2 \left( \frac{1}{\eta ^d}\frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} \right) + \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^c} \frac{P}{q^c} + 2 \alpha \frac{1}{\eta ^d} \frac{P}{Q} \right) \text {det}(H)^{-1} <0 . \end{aligned}$$

Next consider the LCFS:

$$\begin{aligned} \frac{dq^c}{d {\bar{p}}^{\text {cred}}}= & {} \left( (\phi ^c- \sigma ) \left( \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} \right) + (\sigma -\phi ^r) \frac{1}{\eta ^d} \frac{P}{Q} \right) \text {det}(H)^{-1}<0 \\ \frac{dq^r}{d {\bar{p}}^{\text {cred}}}= & {} \left( (\sigma - \phi ^r ) \left( \frac{1}{\xi ^c} \frac{P}{q^c} - \frac{1}{\eta ^d} \frac{P}{Q} \right) - ( \phi ^c-\sigma ) \frac{1}{\eta ^d} \frac{P}{Q} \right) \text {det}(H)^{-1} >0 \\ \frac{d c}{d {\bar{p}}^{\text {cred}}}= & {} \left( (\phi ^c-\sigma )^2 \left( \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^r} \frac{P}{q^r} \right) + (\sigma - \phi ^r)^2 \left( \frac{1}{\eta ^d} \frac{P}{Q} - \frac{1}{\xi ^c} \frac{P}{q^c} \right) + 2(\phi ^c-\sigma ) (\sigma -\phi ^c) \frac{1}{\eta ^d} \frac{P}{Q} \right) \text {det}(H)^{-1} < 0. \end{aligned}$$

Intuition for Inefficiency of Mandates

Mandates and Marginal Abatement Costs

For additional intuition for the second-best nature of fuel mandates relative to a first-best cap-and-trade program, we analyze how the marginal abatement cost per unit of emissions relates to the permit price under a cap-and-trade program. We then compare it to how the marginal abatement cost per unit of emissions relates to the credit price under a fuel mandate.

With a cap-and-trade permit system, the first-order condition for an interior solution for each fuel \(i \in \{ c,r \}\) is given by:

$$\begin{aligned} P- \frac{\partial C^i}{\partial q^i} = \tau \phi ^i , \end{aligned}$$
(B.1)

where \(\tau\) is the permit price.

For each fuel \(i \in \{ c,r \}\), the marginal abatement cost per unit of output is given by \(P- \frac{\partial C^i}{\partial q^i}\) , while the marginal abatement cost \(MAC^i\) per unit of emissions is given by:

$$\begin{aligned} MAC^i = \left( P- \frac{\partial C^i}{\partial q^i} \right) / \phi ^i . \end{aligned}$$
(B.2)

In a cap-and-trade permit system, the marginal abatement cost \(MAC^i\) per unit of emissions given by equation (B.2) equals the permit price \(\tau\) for each fuel i. As a consequence, a cap-and-trade permit system can achieve the first-best when the permit price equals marginal damages: \(\tau = D'(\cdot )\). Under the first-best cap-and-trade permit system, the marginal abatement cost \(MAC^i\) per unit of emissions is equalized across both fuels i and set equal to marginal damages: \(MAC^i = D'(\cdot )\) \(\forall i\).

In contrast, for fuel mandates, the marginal abatement cost \(MAC^i\) per unit of emissions does not equal the credit price \(\lambda\). Formally, for fuel mandates, the analogous first-order conditions are:

$$\begin{aligned}&\left[ q^c: \right] P- \frac{\partial C^c}{\partial q^c} = \lambda \alpha \end{aligned}$$
(B.3)
$$\begin{aligned}&\left[ q^r: \right] P-\frac{\partial C^r}{\partial q^r} = - \lambda \end{aligned}$$
(B.4)

for the RFS, and:

$$\begin{aligned}&\left[ q^c: \right] P- \frac{\partial C^c}{\partial q^c} = \lambda \left( \phi ^c - \sigma \right) \end{aligned}$$
(B.5)
$$\begin{aligned}&\left[ q^r: \right] P-\frac{\partial C^r}{\partial q^r} = - \lambda \left( \sigma -\phi ^r \right) \end{aligned}$$
(B.6)

for the LCFS.

Once again, for each fuel \(i \in \{ c,r \}\), the marginal abatement cost per unit of output is given by \(P- \frac{\partial C^i}{\partial q^i}\) , while the marginal abatement cost \(MAC^i\) per unit of emissions is given by equation (B.2). The fuel mandate credit price \(\lambda\) does not equal the marginal abatement cost per unit of emissions, however. Instead, from the first-order conditions in equations (B.3) and (B.4), the RFS credit price \(\lambda\) is given by the following:

$$\begin{aligned}&\left[ q^c: \right] \left( P- \frac{\partial C^c}{\partial q^c} \right) / \alpha = \lambda \end{aligned}$$
(B.7)
$$\begin{aligned}&\left[ q^r: \right] - \left( P-\frac{\partial C^r}{\partial q^r} \right) = \lambda . \end{aligned}$$
(B.8)

Likewise, from the first-order conditions in Eqs. (B.5) and (B.6), the LCFS credit price \(\lambda\) is given by the following:

$$\begin{aligned}&\left[ q^c: \right] \left( P- \frac{\partial C^c}{\partial q^c} \right) / \left( \phi ^c - \sigma \right) = \lambda \end{aligned}$$
(B.9)
$$\begin{aligned}&\left[ q^r: \right] - \left( P-\frac{\partial C^r}{\partial q^r} \right) / \left( \sigma - \phi ^r \right) = \lambda . \end{aligned}$$
(B.10)

Fuel mandates implicitly tax conventional fuels and subsidize renewable fuels. With the RFS, this is because a firm generates credits from the production of renewable fuel, even though the renewable fuel may have a non-zero emissions factor \(\phi ^r\). With the LCFS, this is because a firm only requires (generates) credits based on how much the carbon intensity \(\phi ^i\) of the fuel \(i=c,r\) exceeds (is lower than) the average carbon intensity \(\sigma\) mandated by the LCFS standard.

Substituting in the marginal abatement cost \(MAC^i\) per unit of emissions given by Eq. (B.2) into Eqs. (B.7)-(B.10) for the fuel mandate credit price \(\lambda\), we get:

$$\begin{aligned}&\left[ q^c: \right] MAC^c = \lambda \alpha / \phi ^c \end{aligned}$$
(B.11)
$$\begin{aligned}&\left[ q^r: \right] MAC^r = - \lambda / \phi ^r \end{aligned}$$
(B.12)

for the RFS, and:

$$\begin{aligned}&\left[ q^c: \right] MAC^c = \lambda \left( \phi ^c - \sigma \right) / \phi ^c \end{aligned}$$
(B.13)
$$\begin{aligned}&\left[ q^r: \right] MAC^r = - \lambda \left( \sigma - \phi ^r \right) / \phi ^r \end{aligned}$$
(B.14)

for the LCFS.

Thus, in contrast to a cap-and-trade program, for fuel mandates, the marginal abatement cost \(MAC^i\) per unit of emissions does not equal the credit price, and is not equalized across fuels i. As a consequence, mandates cannot achieve the first-best. The RFS provides an implicit subsidy for renewable fuels even if the renewable fuel may have a non-zero emissions factor \(\phi ^r\), meaning the RFS does not give an incentive to reduce the emissions from these fuels. Similarly, for fuels with carbon intensities \(\phi ^r\) lower than the average carbon intensity \(\sigma\) mandated by the LCFS, the LCFS provides an implicit subsidy even if the carbon intensity of that fuel is non-zero, meaning the LCFS does not give an incentive to reduce the emissions from these fuels.

Graphical Representation of Second-Best Mandates

Figure 3 illustrates the inefficiency of the mandates graphically. The solid circles are iso-welfare curves that exclude pollution damages. The dashed circles are iso-welfare curves when pollution externalities are internalized.Footnote 33 In the absence of any policy, the competitive market maximizes the sum of consumer and producer surplus at point A, which differs from the social optimum, point B.

To align the competitive and first-best outcome, a regulator can either tax emissions or institute a cap-and-trade program, illustrated in Fig. 3a. Iso-emissions curves are the parallel downward-sloping lines and have slope \((-\phi ^c/\phi ^r)\). The dashed downward-sloping line corresponds to emissions under the no policy outcome. If the government institutes a cap-and-trade program setting the cap at the first-best emission level, represented by the solid downward-sloping line, the competitive market outcome will correspond to the social optimum.

Now consider the efficiency of fuel mandates, illustrated in Fig. 3b. We represent both policies as rays from the origin, where the slope of the ray corresponding to the share of renewable fuel required by the policy. A binding share mandate must pass to the left of the initial share of renewable fuels given by the dashed ray passing through point A. Consider the effect of a binding mandate given by the solid ray. Under the fuel mandate, firms maximize profits at C, resulting in higher renewable and conventional fuel production and higher emissions than the efficient outcome B.

To illustrate that fuel mandates cannot achieve the first-best outcome, suppose the regulator knows the share of renewable fuels or the carbon intensity of fuels under the first-best outcome and sets the mandate at this level, represented by the dotted line through point B. Despite being set at the optimal share, the market maximizes profits at D, away from the first-best outcome. This is due to the subsidy the policies provide for renewable fuel, which reduces the price impact of the policy.

Thus, owing to their implicit subsidy on renewable fuels, whenever unpriced emissions are the sole market failure, fuel mandates are unable to replicate the first-best solution (Helfand 1992; Holland et al. 2009; Lapan and Moschini 2012).

Fig. 3
figure 3

First-best, competitive outcome, and fuel mandates. *Solid circles are iso-welfare curves less damages and the dotted circles are iso-social welfare curves. The parallel downward-sloping lines in a are iso-emission lines with slope \((- \phi ^c/\phi ^r)\). The rays from the origin in b represent fuel mandates

Additional Simulation Results

Figure 4 shows credit purchases in each scenario in panel (b). Credit window purchases decrease as credit window prices become higher, and reach zero when credit window prices exceed the policy constraint’s shadow price.

Fig. 4
figure 4

Compliance credit purchases. The graphs plot the compliance credits purchased on the y-axes against compliance credit prices on the x-axes for different RFS mandated shares \(\alpha\) and different LCFS mandated carbon intensity standards \(\sigma\), respectively

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Lade, G.E., Lin Lawell, CY.C. The Design of Renewable Fuel Mandates and Cost Containment Mechanisms. Environ Resource Econ 79, 213–247 (2021). https://doi.org/10.1007/s10640-021-00558-w

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Keywords

  • Renewable fuels
  • Share mandates
  • Intensity standards
  • Cost containment mechanisms
  • Hybrid policies

Mathematics Subject Classification

  • H23
  • Q42
  • Q54
  • Q58