Intertemporal Emission Permits Trading Under Uncertainty and Irreversibility
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Abstract
This paper analyzes the effect of emission permit banking on clean technology investment and abatement under conditions where the stringency of the future cap is uncertain. We examine the problem of heterogeneous firms minimizing the cost of intertemporal emission control in the presence of stochastic future pollution standards and emission permits that are tradable across firms and through time. A firm can invest in clean capital (an improved pollution abatement technology) to reduce its abatement cost. We consider two possibilities: that investment is reversible or irreversible. Uncertainty is captured within a two period model: only the current period cap is known. We show that if banking is positive and marginal abatement costs are sufficiently convex, there will be more abatement and investment in clean technology under uncertainty than there would be under certainty and no banking. These results are at odds with the common belief that uncertainty on future environmental policy is a barrier to investment in clean capital. Moreover, under uncertainty and irreversibility, we find that there are cases where banking enables firms to invest more in clean capital.
Keywords
Cap uncertainty Abatement Banking Investment in clean capital Irreversibility1 Introduction
The ultimate goal of climate policy is to stabilize greenhouse gas concentrations at a level that is sustainable in the Brundtland 1987 sense. However, determining this level is difficult due to uncertainty within the geophysical and ecological sciences, in the cost of decarbonizing economies (Heal and Kriström 2002) as well as in policy acceptability. Uncertainty about midcentury \(\hbox {CO}_{2}\) emissions target is likely to affect firms’ current abatement as well as their current technological choices, even more when considering the irreversible nature of such choices: a utility company will find it too expensive to remove a scrubber (and recover the cost) if emissions standards are less stringent than expected. However, an important characteristic of the emission permit market design provides some flexibility for the firm: it allows permits to be traded across compliance periods. This paper considers intertemporal emission trading under uncertainty over future standards and technological irreversibility. It explores the consequences of uncertainty and irreversibility for technological investment and current and future abatement. As a result, we are able to make policy recommendations to avoid the usual adverse effect of uncertainty on irreversible technological choices.
Our paper is related to the literature analyzing intertemporal emissions permits. Even if theoretical analysis of intertemporal emission permit trading has received considerable interest (Kling and Rubin 1997; Rubin 1996; Cronshaw and Kruse 1996; Strandlund et al. 2005; Seifert et al. 2008; Slechten 2013), papers that explicitly focus on uncertainty have appeared more recently and are less abundant. Among previous investigations of intertemporal permit trading under uncertainty, Schennach (2000) examined the implications of uncertainty on emission permit trading. In the context of Title IV of the US Clean Air Act Amendments (1990), electricitygenerating units face uncertainty in their marginal abatement cost, in the regulatory environment, and in the demand for electricity. Schennach suggests that the expected allowance price path can rise at a rate less than the discount rate even when the bank is not expected to be depleted (i.e. expected banking is strictly positive). This result arises from her assumption that there is a nonnegativity constraint on the bank of allowances (only banking is allowed, not borrowing). Feng and Zhao (2006) discuss the effects of abatement costs uncertainty and conclude that more permits will be banked when the expected marginal value of permits rises. Also focusing on abatement cost uncertainty, Phaneuf and Requate (2002) show that banking generates an incentive for firms to delay investment in improved abatement technology. Laffont and Tirole (1996) find that a standalone spot market for pollution permits induces too much investment in pollution abatement, while the introduction of a future market reduces this incentive but is not the optimal way to control pollution. Finally, Zhang (2007) studies the effect of uncertainty on electricity producing firms. Uncertainty affects future electricity prices and is then conveyed to future permit prices. Firms emit less in volatile markets than they would if future market conditions were known; as a consequence, the industry as a whole will accumulate more permits.
Our paper is also related to the literature on irreversible investment under uncertainty. Introducing technological choice, Zhao (2003) describes how abatement cost uncertainty affects an irreversible technological investment. In the spirit of Arrow and Fisher (1974), Henry (1974) or Kolstad (1996), increased uncertainty reduces investment for risk neutral firms in partial equilibrium by generating an incentive to wait before investing. Turning to industrywide uncertainty this result may not be robust. The argument runs as follows: if one firm waits, other firms may invest and consequently drive down the price, making further investment suboptimal. Nevertheless, the author shows that both industry and firmspecific uncertainty reduce the investment incentive at equilibrium, i.e., the typical adverse effect of uncertainty on irreversible technological choice prevails. However, Zhao does not account for the counteracting flexibility that arises from intertemporal trading on the permits market.
The principal aim of banking of allowances is to enable the phasein of a trading program (see Schennach 2000) and a key characteristic of this phasein is that the next phase standards are not perfectly known. However, the literature focusing on future standards uncertainty is quite recent. The idea of a trading program with potentially more stringent regulations following an initial, less restrictive compliance period can be captured by a twoperiod model that represents actions taken early on and actions performed in the second period when uncertainty has been resolved. DurandLasserve et al. (2010), use an applied general equilibrium model, to consider a “hard cap” scenario and a “soft cap” scenario for the end of 2020. They show that a higher hard cap probability leads to more abatement and more banking until 2020. Turning to a more theoretical approach, Fischer and Sterner (2012) find that future cap uncertainty will affect current abatement and R&D investment depending on the shape of the cumulative marginal abatement cost curve that provides a measure for prudence. R&D, by changing this shape, interacts with prudence. Therefore the latter paper focuses on a Jensen effect of uncertainty but does not account for the joined effect of technological irreversibility and future cap uncertainty. However, such an effect of irreversibility is worth considering since it could potentially be mitigated with the flexibility provided by intertemporal permit trading.
This paper fills a theoretical gap within the environmental literature by analyzing the effect on abatement and irreversible technological choices of uncertainty on future permit allowances. The objective is first to analyze the effect of uncertainty on clean capital investment in the presence of pollution permit markets and banking. Second, it is to examine whether under uncertainty, banking provides an incentive for investing in clean capital. Based on this assessment, we provide policy recommendation on the desirability of banking of allowances. We explore the problem of heterogenous firms minimizing the cost of intertemporal emission control in the presence of an uncertain future cap and emission permits that are tradable across firms and through time. A firm can invest in clean capital (an improved pollution abatement technology) to reduce its abatement cost. Uncertainty is captured within a two period model. Only the current period cap is known by the firms. We find that if banking is positive and marginal abatement costs are sufficiently convex, there will be more abatement and investment in clean technology under uncertainty than there would be under certainty and no banking. Accounting for the irreversibility of investment in clean technology, additional parameters that are crucial for the effect of uncertainty and of banking are those describing uncertainty and the discount rate. It is possible to find values for these parameters such that uncertainty (and irreversibility) induces more abatement and clean investment than when uncertainty is ignored. Moreover, in the most plausible case where irreversibility is only binding when the cap is increased, allowing for positive banking induces more investment.
The general model setup is presented in the next section. In Sect. 3, the model is solved under certainty as a benchmark. Uncertainty on the future cap is introduced in Sect. 4. Section 5 concludes.
2 Model SetUp
The regulator allows trading on a permit market and banking between the two periods. Firms are risk neutral. Firm i chooses abatement level \(Q_{ti}\), \(t=1,2,\) investment level in technology that allows increasing the stock \(K_{ti}\), and level of permits banking \(B_{1i}\) in period 1, given that the future cap and therefore the price of permits is uncertain. The investment cost function is linear in the investment level, with the unit price of the capital good given by \(k>0.\)
The solution under a certain second period cap is derived in the next section, as a benchmark against which the joined effect of uncertainty and irreversibility will be assessed in the following section.
3 Benchmark: The Future Cap is Known with Certainty
We now derive the optimal individual and aggregate levels of abatement and investment in clean capital in the benchmark case where the cap at period 2 is known with certainty.
Each firm adjusts its stream of emissions and chooses its investment in clean capital to minimize the cost of compliance.
3.1 No or Nonbinding Irreversibility
FOCs when the second period cap is certain

3.2 Binding Irreversibility
3.3 The Role of Banking
However, when there is irreversibility, we show that \(K_{1}^{*\text {irrwb}}>K_{1}^{*\text {irr}}\) (see “Deterministic Case with Binding Irreversibility” section of Appendix 2). Under irreversibility, first period investment is aimed not at reducing first period abatement cost only, but at reducing the discounted sum of abatement costs over the two periods. Positive banking still provides the opportunity for larger abatement in period 1 with the counterpart of lower abatement in period 2. This generates a lower discounted sum of abatement costs over the two periods and therefore a lower need for investment in first period. With binding irreversibility, positive banking and investment in clean capital are substitutes, in the sense that positive banking leads to less investment in clean capital at period 1.
The following proposition summarizes the abatement, banking and investment behaviors of firms in the absence of uncertainty on the second period cap.
Proposition 1
 1.
Under the assumption that caps at the two periods are lower than BAU emissions, first period abatement and investment in clean capital are always positive. When banking permits is allowed, it is positive for a low interest rate relative to the change in the lenience of the cap.
 2.
Banking, abatement and investment in clean capital at period 1 are smaller when irreversibility is accounted for and binding, which occurs when \(r<r^{*}\) defined by \((1+r^{*})r^{*\alpha 1}=1\).
 3.
Positive banking and investment in clean capital at period 1 are complements when irreversibility is not binding, whereas they are substitutes when irreversibility is binding.
We now turn to the optimal behavior of firms when the second period cap is uncertain.
4 The Future Cap is Uncertain
4.1 No or Nonbinding Irreversibility
FOCs when the second period cap is uncertain

To assess the effect of uncertainty, we compare abatement levels with and without uncertainty. We obtain that uncertainty on the future cap may induce either more or less abatement than when the future cap is certain, depending on the characteristics of the optimized marginal abatement cost function. More precisely, \(Q_{1}^{\sharp }>Q_{1}^{*}\) and \(B_{1}^{\sharp }>B_{1}^{*}\) if and only if \(\alpha >2(1+\beta )\). The argument runs as follows. Whether or not there is uncertainty, the marginal benefit of investment in clean capital is equal to its marginal cost, which is constant. With our specification of the abatement cost function, the fact that \(C_{K}(Q,K)\) is constant implies that K is proportional to \(Q^{\frac{\alpha }{1+\beta }},\) which in turn implies that \(C_{Q}(Q,K)\) is proportional to \(Q^{\frac{\alpha 1\beta }{1+\beta }}.\) This means that the optimized marginal abatement cost \(C_{Q}(Q)\) is a convex function of Q when \(\frac{\alpha 1\beta }{1+\beta }>1\) i.e. \(\alpha >2(1+\beta ),\) and a concave function of Q when \(\alpha <2(1+\beta ).\) Therefore the convexity of the marginal cost curve depends on both parameters \(\alpha \) and \(\beta \).
Results are summarized in the following proposition, where we focus on the case \(\alpha >2(1+\beta )\) in which the results do not conform to the common belief that uncertainty on the future cap is detrimental to investment in clean capital.
Proposition 2
 1.
there is more abatement and more investment in clean capital than when the second period cap is certain if and only if \(\alpha >2(1+\beta )\);
 2.
positive banking and investment in clean capital at period 1 are complements.
4.2 Binding Irreversiblility
We now consider that investment in clean capital is fully irreversible. A firm cannot disinvest at period 2 if it has too much capital compared with what would be optimal. The most favorable case for irreversibility to be binding occurs when \({\widetilde{A}}_{2i}=A_{1i}+{\underline{\Delta }}_{i}\). We consider two a priori possible cases: (a) irreversibility is binding only when the cap is increased and (b) irreversibility is binding whatever the second period cap. In each case we proceed in two steps. We consider first that irreversibility is binding and compute the optimal solution for the two periods. Then we establish the conditions under which irreversibility is actually binding.
4.2.1 Irreversibility only Binding when the Cap is Increased
4.2.2 Irreversibility Binding Regardless of the Second Period Cap
4.2.3 Irreversibility Frontiers

The first irreversibility frontier separates the cases where irreversibility is not binding and where it is binding only when the cap is increased. It is characterized by \({\underline{K}}_{2}^{\sharp }=K_{1}^{\sharp }\) and can be expressed analytically as \(r^{\sharp a}(q)\) (see “Stochastic Case with Binding Irreversibility” section of Appendix 2). We show that \(r^{\sharp a}(1)=r^{*},\) and that \(r^{\sharp a}(q)\) is a decreasing function of q. Irreversibility is binding for \(r<r^{\sharp a}(q)\).

The second irreversibility frontier separates the cases where irreversibility is binding only when the cap is increased and where it is binding regardless of the cap. It is characterized by \({\overline{K}} _{2}^{\sharp \text {ira}}=K_{1}^{\sharp \text {ira}}\). Notice (see “Stochastic Case with Binding Irreversibility” section of Appendix 2) that this equation, valid on the second irreversibility frontier, is identical to the equation which implicitly gives \(Q_{1}^{\sharp \text {irb}}\). Hence \(Q_{1}^{\sharp \text {ira}}=Q_{1}^{\sharp \text {irb}}\) and \(K_{1}^{\sharp \text {ira}}=K_{1}^{\sharp \text {irb}}\) (also clear from a continuity argument) on the irreversibility frontier. We cannot specify the equation of this frontier \(r^{\sharp b}(q)\) further and we turn to numerical simulations to characterize it more precisely.
4.2.4 The Role of Banking
When irreversibility is binding regardless of the cap, the argument for the effect of banking on first period investment in clean capital runs as it does without uncertainty. Banking and first period investment in clean capital are substitutes (see “Stochastic Case with Binding Irreversibility” section of Appendix 2).
When irreversibility is only binding when the cap is increased, firms may invest in clean capital at period 2 if the cap happens to be strengthened. That is, first period clean capital is the only capital installed for the two periods (and there is an abatement cost reduction in the second period) only if the cap is increased, which occurs with a probability \(q<1\). As a result, the discounted sum of abatement costs over the two periods is higher, and therefore there is a higher need for investment in first period. Banking and first period investment are complements (see “Stochastic Case with Binding Irreversibility” section of Appendix 2).
The following proposition summarizes the results.
Proposition 3
 1.When the second period cap is uncertain and irreversibility is binding regardless of the second period cap:

abatement is larger than when the second period cap is certain if and only if \(\alpha >2\);

it is possible to find values for q and r such that there is more investment in clean capital than when the second period cap is certain;

positive banking and investment in clean capital are substitutes.

 2.When the second period cap is uncertain and irreversibility is only binding for a high cap:

it is possible to find values for q and r such that there is more investment in clean capital than when the second period cap is certain;

positive banking and investment in clean capital are complements.

We find therefore that banking may be a means to take advantage of uncertainty and irreversibility in the most plausible case where irreversibility is only binding when the cap is increased, because it breaks the substitutability between banking and earlier investment in clean capital. Note that on the contrary, Phaneuf and Requate (2002) find that banking and investment in clean capital are always substitutes since they do not allow for any investment after the resolution of uncertainty.
5 Conclusion
This paper considers intertemporal emission permit trading under future cap uncertainty and technological irreversibility. In particular, it explores the consequences of uncertainty and irreversibility on investment in clean capital, and current and future abatements. We show that banking can be an effective means to take advantage of uncertainty and investment irreversibility. A necessary condition for a positive effect of banking on clean capital investment is the convexity of marginal abatement costs. Therefore, we claim for a precise information on the shape of these marginal abatement costs that would be a requirement before any policy recommendation concerning the banking of allowances could be made.
We acknowledge that we impose a large number of simplifying assumptions in our model: firms’ production decisions are ignored, micro abatement costs are such that they allow for exact aggregation. Such a simple framework cannot be used to derive general conclusions but is sufficient to conclude that uncertainty and irreversiblity do not always result in less technology adoption and less abatement.
An extension of this work would be to endogenize the regulator’s optimal second period cap. It would make sense that the second period cap depends on the level at which firms invested in first period. This would give rise to an interesting holdup problem since firms would foresee the relationship between the first period investment and second period cap.
Footnotes
 1.
These effects would qualitatively remain the same under an infinite horizon assumption, but models of irreversible investment under uncertainty can quickly become intractable under such a more general assumption.
 2.
Not necessarily in an optimal way. The cap can result from an international agreement.
 3.
That is, we do not specify how it is decided that firm i will get \(A_{1i}\) and \(A_{2i}\).
 4.
Recall that \(C_{QQ}(Q_{1})>0.\)
 5.
See “Stochastic Case with Binding Irreversibility” section of Appendix 2.
 6.
Recall that \(C_{QQ}(Q_{1})>0\)
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