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Environmental and Resource Economics

, Volume 71, Issue 1, pp 73–97 | Cite as

Intertemporal Emission Permits Trading Under Uncertainty and Irreversibility

  • Aude Pommeret
  • Katheline SchubertEmail author
Article

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 Irreversibility 

1 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 mid-century \(\hbox {CO}_{2}\) 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), electricity-generating 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 non-negativity 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 stand-alone 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 industry-wide 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 firm-specific 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 phase-in of a trading program (see Schennach 2000) and a key characteristic of this phase-in 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 two-period model that represents actions taken early on and actions performed in the second period when uncertainty has been resolved. Durand-Lasserve 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 set-up 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 Set-Up

We focus on a two-period framework that properly takes into account the effects of uncertainty and irreversibility on interest variables.1 We consider a tradable emission permit market consisting of N price taking firms with rational expectations about permit prices. Firm production decisions are ignored, in order to focus on emission permit trading. The abatement cost of firm i is \(C_{i}(Q_{i},K_{i}),\) where \(Q_{i}\) is the abatement level and \(K_{i}\) the stock of abatement capital or technology (hereafter referred to as clean capital). Investments that increase this specific form of capital are assumed to be difficult to reverse: for example, a utility company will find it costly to remove (and recover the cost of) a scrubber it has installed. We make the extreme assumption that investment is fully irreversible. By allowing the abatement cost to depend on i we account for the heterogeneity of the firms, a major advantage for tradable permits. However, in order to derive aggregate results, we choose a specification of the individual cost function permitting an exact aggregation, such that the aggregate cost function is the envelope of the individual cost functions. We denote the aggregate abatement cost function by C(QK),  with \(Q=\sum \nolimits _{i=1}^{N}Q_{i}\) and \(K=\sum \nolimits _{i=1}^{N}K_{i}\). The individual abatement cost function is specified as follows:
$$\begin{aligned} C_{i}(Q_{i},K_{i})=c_{i}Q_{i}^{\alpha }K_{i}^{-\beta },\quad c_{i}>0,\text { }\beta>0,\text { }\alpha >1+\beta . \end{aligned}$$
(1)
Therefore it exhibits the standard characteristics: \(C(0,K)=0,C_{iQ}>0,C_{iQQ}>0\,\)and \(C_{iK}<0,C_{iKK}>0.\) Assumption \(\alpha >1+\beta \) ensures the convexity of the cost function. Note that \(\alpha \) and \(\beta \) are the same across firms, whereas the constant multiplicative term differs. It can easily be shown (see Appendix 1) that with specification (1), aggregation yields:
$$\begin{aligned} C(Q,K)=cQ^{\alpha }K^{-\beta }\text { with }c=\left( \sum _{i=1}^{N}c_{i} ^{\frac{1}{(1+\beta )-\alpha }}\right) ^{(1+\beta )-\alpha }, \end{aligned}$$
(2)
and that the repartition of the abatement and investment efforts among firms obeys the following rule:
$$\begin{aligned} Q_{i}&=\left( \frac{c}{c_{i}}\right) ^{\frac{1}{\alpha -(1+\beta )} }Q, \end{aligned}$$
(3)
$$\begin{aligned} K_{i}&=\left( \frac{c}{c_{i}}\right) ^{\frac{1}{\alpha -(1+\beta )}}K. \end{aligned}$$
(4)
Business as usual (BAU) emissions of firm i are supposed to be the same at each period, and are denoted \({\overline{U}}_{i}.\) Aggregate BAU emissions are \({\overline{U}}=\sum \nolimits _{i=1}^{N}{\overline{U}}_{i}\). The first period is the current period. The regulator sets a cap2 on polluting emissions \(A_{1}<{\overline{U}},\) and allocates permits to the firms free of charge, according to an unspecified allocation scheme.3 Firm i receives \(A_{1i},\) with \(A_{1}=\sum \nolimits _{i=1}^{N}A_{1i}.\) Uncertainty affects the second period cap \({\widetilde{A}}_{2}\). With a probability q the cap is relaxed compared to period 1 and becomes \(A_{1}+{\underline{\Delta }}\), \({\underline{\Delta }}>0,\) and with a probability \((1-q)\) environmental policy is strengthened and the cap becomes \(A_{1}-{\overline{\Delta }}\), \({\overline{\Delta }}>0\). We assume that even when the cap is relaxed, it is not high enough to cover BAU emissions: \(A_{1}+{\underline{\Delta }}<{\overline{U}}\). The permit allocation mechanism among firms remains the same as in the first period, and firm i gets
$$\begin{aligned} {\widetilde{A}}_{2i}=\left\{ \begin{array}[c]{ll} A_{1i}+{\underline{\Delta }}_{i} &{} (q)\\ A_{1i}-{\overline{\Delta }}_{i} &{} (1-q) \end{array}\right. {,} \quad i=1,\ldots ,N \end{aligned}$$
(5)
where \({\underline{\Delta }}_{i}\) and \({\overline{\Delta }}_{i}\) are known by the firm, and with \(\underline{\Delta }=\sum \nolimits _{i=1}^{N}{\underline{\Delta }}_{i}\) and \({\overline{\Delta }}=\sum \nolimits _{i=1}^{N}{\overline{\Delta }}_{i}.\) In order to easily capture the effect of uncertainty, we consider a mean preserving spread uncertainty: \(A_{2}=E_{1}\left[ {\widetilde{A}}_{2}\right] =A_{1} +q{\underline{\Delta }}-(1-q){\overline{\Delta }}\).

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.

Abatements \(Q_{1i}\) and \(Q_{2i}\), permit purchases \(y_{1i}\) and \(y_{2i}\) at the two periods, and first period banking \(B_{1i}\) must satisfy:
$$\begin{aligned} y_{1i}&=B_{1i}+{\overline{U}}_{i}-A_{1i}-Q_{1i}, \end{aligned}$$
(6)
$$\begin{aligned} y_{2i}&=-B_{1i}+{\overline{U}}_{i}-A_{2i}-Q_{2i}\quad \forall i=1,\ldots ,N. \end{aligned}$$
(7)
Note that permit banking, as opposed to contemporaneous trading among agents, changes the nature of the caps since all firms can emit above the cap at some point in time. What is required is that firms satisfy the cumulative cap over the two periods. Summing Eqs. (6) and (7) over i yields the permit market equilibrium conditions:
$$\begin{aligned} B_{1}+{\overline{U}}-A_{1}-Q_{1}&=0, \end{aligned}$$
(8)
$$\begin{aligned} -B_{1}+{\overline{U}}-A_{2}-Q_{2}&=0. \end{aligned}$$
(9)
Total abatement over the two periods is:
$$\begin{aligned} Q_{1}+Q_{2}=2{\overline{U}}-(A_{1}+A_{2})=2({\overline{U}}-A_{1})+\overline{\Delta }-q({\underline{\Delta }}+{\overline{\Delta }})=\Gamma (q). \end{aligned}$$
(10)
As we have made the assumptions that the cap set in period 1 is lower than BAU emissions (\(A_{1}<{\overline{U}}\)) and that is the case for the cap set in period 2 as well, even when the latter is relaxed (\(A_{1}+{\underline{\Delta }}<{\overline{U}}\)), total abatement is positive: \(\Gamma (q)>0.\)

Each firm adjusts its stream of emissions and chooses its investment in clean capital to minimize the cost of compliance.

3.1 No or Non-binding Irreversibility

Ignoring in a first stage irreversibility of the investment in clean capital, optimal abatement, banking and investment decisions of firm i are given by the solution to the following problem:
$$\begin{aligned} V_{1i}(K_{0i},A_{1i})= & {} \min _{Q_{1i},I_{1i},B_{1i}}\left\{ C_{i}(Q_{1i}, K_{0i}+I_{1i})+kI_{1i}-p_{1}\left[ A_{1i}-\left( {\overline{U}}_{i} -Q_{1i}\right) -B_{1i}\right] \right. \nonumber \\&\left. +\,\frac{1}{1+r}V_{2i}\left( K_{1i},A_{2i}\right) \right\} , \end{aligned}$$
(11)
where \(I_{1i}=K_{1i}-K_{0i}\) is the investment in clean capital at period 1 and \(p_{1}\) the permit price at this same period. \(K_{0i}\), the initial clean capital stock of firm i, and r, the discount rate, are given with \(r<1\). This program can be solved using backward induction. Therefore, we first focus on the second period problem and then turn to that of the first period. The second period program is:
$$\begin{aligned} V_{2i}(K_{1i},A_{2i})=\min _{Q_{2i},I_{2i}}\left\{ C_{i}(Q_{2i},K_{1i} +I_{2i})+kI_{2i}-p_{2}\left[ B_{1i}-{\overline{U}}_{i}+A_{2i}+Q_{2i}\right] \right\} . \quad \quad \end{aligned}$$
(12)
where \(I_{2i}=K_{2i}-K_{1i}\) is the investment in clean capital and \(p_{2}\) the permit price at period 2.
The first order optimality conditions at periods 1 and 2 state that the marginal cost of abatement should equal the permit price, that the marginal reduction in abatement costs due to investment in clean capital should be equal to the unit cost of investment, and that permit prices at the two periods obey the Hotelling rule. They are listed in Table 1. In what follows, we use the fact that with our specification of the individual cost functions, aggregate marginal abatement cost and investment benefit, \(C_{Q}(Q,K)\) and \(C_{K}(Q,K),\) are equal to their individual counterparts, \(C_{iQ}(Q_{i},K_{i})\) and \(C_{iK}(Q_{i},K_{i}),\) \(\forall i\) (see Appendix 1), and we directly consider aggregate marginal costs.
Table 1

FOCs when the second period cap is certain

Resolutions of first and second period programs are given in “Deterministic Case Without Binding Irreversibility” section of Appendix 2. Using the first order conditions and the permit market equilibrium conditions allows us to obtain optimal aggregate abatement, investment and banking at period 1:
$$\begin{aligned} Q_{1}^{*}&=\frac{\Gamma (q)}{1+f(r)}\text { with }f(r)=\left[ r^{\beta }(1+r)\right] ^{\frac{1}{\alpha -1-\beta }}, \end{aligned}$$
(13)
$$\begin{aligned} K_{1}^{*}&=\left[ \frac{1+r}{r}\frac{c\beta }{k}\left. Q_{1}^{*}\right. ^{\alpha }\right] ^{\frac{1}{1+\beta }}, \end{aligned}$$
(14)
$$\begin{aligned} B_{1}^{*}&=Q_{1}^{*}-\left( {\overline{U}}-A_{1}\right) =\frac{\left( {\overline{U}}-A_{2}\right) -f(r)\left( {\overline{U}} -A_{1}\right) }{1+f(r)}. \end{aligned}$$
(15)
As \(\Gamma (q)>0,\) first period abatement is always strictly positive. Banking may be positive or negative, the necessary and sufficient condition to have positive banking in the first period being:
$$\begin{aligned} B_{1}^{*}>0\Leftrightarrow f(r)<\frac{{\overline{U}}-A_{2}}{{\overline{U}}-A_{1}}. \end{aligned}$$
(16)
Condition (16) can be interpreted as follows: positive banking occurs when the discount rate is not too high relative to the change in the lenience of the cap across the two periods. Also note that:
$$\begin{aligned} \frac{\partial B_{1}^{*}}{\partial r}<0, \; \frac{\partial B_{1}^{*}}{\partial \left( {\overline{U}}-A_{1}\right) }<0, \; \frac{\partial B_{1}^{*}}{\partial \left( {\overline{U}}-A_{2}\right) }>0. \end{aligned}$$
In particular, the higher the discount rate, the less concerned the firm is about future costs as compared to present costs, and the weaker the incentive to bank permits. In addition, the more lenient the first (resp. second) period cap, the larger (resp. smaller) the banking. Consistent with the intuition, banking at period 1 can be strictly positive, but is less than the second period cap reduction (\(B_{1}={\overline{U}}-A_{2}-Q_{2}\) with \(Q_{2}^{*}>0)\), because firms, knowing that the cap will be reduced in the next period, spread the abatement effort over the two periods.

3.2 Binding Irreversibility

In a second stage we consider full irreversibility of investment in clean capital: investment at each period must be positive. We assume that the initial clean capital \(K_{0}\) is small enough for investment \(I_{1}^{*}=K_{1}^{*}-K_{0}\) to be strictly positive in the first period. We show that second period investment \(I_{2}=K_{2}^{*}-K_{1}^{*}\) is positive if and only if \((1+r)r^{\alpha -1}\ge 1\) (see “Deterministic Case with Binding Irreversibility” section of Appendix 2). There exists a threshold value for the interest rate, denoted \(r^{*}\) and defined by:
$$\begin{aligned} (1+r^{*})r^{*\alpha -1}=1, \end{aligned}$$
(17)
above which investment in period 2 is positive, and below which it is nil because of irreversibility. Equation (17) is the equation for the irreversibility frontier. If \(r\ge r^{*},\) whether or not there is irreversibility does not alter optimal decisions. If \(r<r^{*}\), irreversibility is binding and optimal decisions are modified.
When irreversibility is binding, the second period value of firm i becomes:
$$\begin{aligned} V_{2i}^{\text {irr}}(K_{1i},A_{2i})=\min _{Q_{2i}}\left\{ C_{i}(Q_{2i},K_{1i})-p_{2}\left[ B_{1i}-{\overline{U}}_{i}+A_{2i}+Q_{2i}\right] \right\} . \end{aligned}$$
(18)
We proceed as above to obtain optimal abatement, banking and aggregate clean capital at period 1 (see “Deterministic Case with Binding Irreversibility” section of Appendix 2 and Table 1), and obtain:
$$\begin{aligned} Q_{1}^{*\text {irr}}&=\frac{\Gamma (q)}{1+g(r)}\text { with } g(r)=(1+r)^{\frac{1}{\alpha -1}}, \end{aligned}$$
(19)
$$\begin{aligned} K_{1}^{*\text {irr}}&=\left[ \left( 1+g(r)\right) \frac{c\beta }{k}\left. Q_{1}^{*\text {irr}}\right. ^{\alpha }\right] ^{\frac{1}{1+\beta }}, \end{aligned}$$
(20)
$$\begin{aligned} B_{1}^{*\text {irr}}&=Q_{1}^{*\text {irr}}-\left( {\overline{U}} -A_{1}\right) =\frac{\left( {\overline{U}}-A_{2}\right) -g(r)\left( {\overline{U}}-A_{1}\right) }{1+g(r)}. \end{aligned}$$
(21)
The necessary and sufficient condition for positive banking in the first period is now:
$$\begin{aligned} B_{1}^{*\text {irr}}>0\Leftrightarrow g(r)<\frac{{\overline{U}}-A_{2} }{{\overline{U}}-A_{1}}. \end{aligned}$$
(22)
Again, positive banking occurs when the discount rate is not too high relative to the change in the lenience of the cap across the two periods. We still have:
$$\begin{aligned} \frac{\partial B_{1}^{*\text {irr}}}{\partial r}<0, \; \frac{\partial B_{1}^{*\text {irr}}}{\partial \left( {\overline{U}} -A_{1}\right) }<0,\;\frac{\partial B_{1}^{*\text {irr}}}{\partial \left( {\overline{U}}-A_{2}\right) }>0. \end{aligned}$$
With binding irreversibility, for a given discount rate, we easily see that first period abatement is smaller than under reversibility:
$$\begin{aligned} Q_{1}^{*\text {irr}}<Q_{1}^{*}\Leftrightarrow g(r)>f(r)\Leftrightarrow r^{\alpha -1}(1+r)<1\Leftrightarrow r<r^{*}, \end{aligned}$$
which is precisely the condition that ensures irreversibility is binding. It follows that investment in clean capital and banking are also smaller. A corollary is that the necessary and sufficient condition for positive banking is more stringent when irreversibility is binding.

3.3 The Role of Banking

Without banking, aggregate abatement is equal at each period to what is needed to comply with the cap, irrespective of irreversibility: \(Q_{1}^{\text {wb} }={\overline{U}}-A_{1}\), and \(Q_{2}^{\text {wb}}={\overline{U}}-A_{2}.\) Investment takes place at both periods to decrease the abatement cost, up to the point where the marginal benefit of investment equals its marginal cost. We obtain the following optimal levels of clean capital respectively without and with irreversibility (see “Deterministic Case with Binding Irreversibility” section of Appendix 2):
$$\begin{aligned} K_{1}^{*\text {wb}}&=\left[ \frac{1+r}{r}\frac{c\beta }{k}\left. Q_{1}^{\text {wb}}\right. ^{\alpha }\right] ^{\frac{1}{1+\beta }} , \end{aligned}$$
(23)
$$\begin{aligned} K_{1}^{*\text {irr-wb}}&=\left[ \frac{c\beta }{k}\left( \left. Q_{1}^{\text {wb}}\right. ^{\alpha }+\frac{1}{1+r}\left. Q_{2}^{\text {wb} }\right. ^{\alpha }\right) \right] ^{\frac{1}{1+\beta }}. \end{aligned}$$
(24)
In the absence of irreversibility, if positive (and only positive) banking is allowed, \(K_{1}^{*\text {wb}}<K_{1}^{*}\) (see Eqs. (14) and (23)). Positive banking provides the opportunity for larger abatement in period 1 with the counterpart of lower abatement in period 2. This translates into more first period investment in clean capital as well, in order to reduce first period abatement cost. Positive banking and investment in clean capital are complements, in the sense that positive banking leads to more investment in clean capital at period 1.

However, when there is irreversibility, we show that \(K_{1}^{*\text {irr-wb}}>K_{1}^{*\text {irr}}\) (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

Absent uncertainty:
  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. 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. 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 Non-binding Irreversibility

As in the benchmark above, we first assume that investment is fully reversible. If a firm happens to have a stock of clean capital at period 2 that is higher than the optimal level, it can sell some capital at price k. Firm i is risk neutral and takes each period permit prices \(p_{1}\) and \(p_{2}\) as given. It seeks to minimize the present value of its costs subject to the condition on the cumulative cap over the two periods. The optimal value function encompasses current abatement and investment costs, current permit purchase and the present value of future costs. Firm i chooses abatement level \(Q_{ti}\), \(i=1,2,\) investment in clean capital \(I_{ti}\), and the level of banking \(B_{1i},\) given that permit prices are uncertain. Its program in period 1 is therefore:
$$\begin{aligned} V_{1i}(K_{0i},A_{1i}) =&\min _{Q_{1i},I_{1i},B_{1i}}\left\{ C_{i} (Q_{1i},K_{0i}+I_{1i})+kI_{1i}-p_{1}\left[ A_{1i}-({\overline{U}}_{i} -Q_{1i})-B_{1i}\right] \right. \nonumber \\&\left. +\,\frac{1}{1+r}E_{1}\left[ V_{2i}(K_{1i},{\widetilde{A}} _{2i})\right] \right\} , \end{aligned}$$
(25)
\({\widetilde{A}}_{2i}\) being defined by Eq. (5), and with:
$$\begin{aligned} E_{1}\left[ V_{2i}(K_{1i},{\widetilde{A}}_{2i})\right] =qV_{2i}(K_{1i} ,A_{1i}+{\underline{\Delta }}_{i})+(1-q)V_{2i}(K_{1i},A_{1i}-\overline{\Delta }_{i}). \end{aligned}$$
(26)
Again, we start by solving the second period program.
If the cap is increased in period 2 (lower-bar notation denotes this case), firm i abates \({\underline{Q}}_{2i}^{\sharp }\), invests \(({\underline{K}} _{2i}^{\sharp }-K_{1i}^{\sharp })\) and purchases permits at a price \(\underline{p}_{2}\). On the contrary if environmental policy happens to be more stringent (upper-bar notation denotes this case), firm i abates \({\overline{Q}}_{2i}^{\sharp },\) invests \(({\overline{K}}_{2i}^{\sharp } -K_{1i}^{\sharp })\) and purchases permits at a price \({\overline{p}}_{2}\) at period 2. Firm \(i^{\prime }s\) programs at period 2 then read:
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}+{\underline{\Delta }}_{i}) =&\min _{{\underline{Q}} _{2i},{\underline{K}}_{2i}}\left\{ C_{i}({\underline{Q}}_{2i},{\underline{K}} _{2i})+k({\underline{K}}_{2i}-K_{1i})\right. \nonumber \\&\left. -\,{\underline{p}}_{2}\left[ B_{1i} +(A_{1i}+{\underline{\Delta }}_{i})-({\overline{U}}_{i}-{\underline{Q}}_{2i})\right] \right\} , \end{aligned}$$
(27)
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}-{\overline{\Delta }}_{i}) =&\min _{{\overline{Q}} _{2i},{\overline{K}}_{2i}}\left\{ C_{i}({\overline{Q}}_{2i},{\overline{K}} _{2i})+k({\overline{K}}_{2i}-K_{1i})\right. \nonumber \\&\left. -\,{\overline{p}}_{2}\left[ B_{1i} +(A_{1i}-{\overline{\Delta }}_{i})-({\overline{U}}_{i}-{\overline{Q}}_{2i})\right] \right\} . \end{aligned}$$
(28)
First order conditions are listed in Table 2.
Table 2

FOCs when the second period cap is uncertain

Resolutions of first and second period programs are given in “Stochastic Case Without Binding Irreversibility” section of Appendix 2. With \(Q_{1}^{\sharp },\) \(K_{1}^{\sharp }\) and \(B_{1}^{\sharp }\) the optimal aggregate abatement, clean capital and banking at period 1, we obtain at the aggregate level:
$$\begin{aligned} f(r)Q_{1}^{\sharp }&=\left( q\left. {\underline{Q}}_{2}^{\sharp }\right. ^{\frac{\alpha -1-\beta }{1+\beta }}+(1-q)\left. {\overline{Q}}_{2}^{\sharp }\right. ^{\frac{\alpha -1-\beta }{1+\beta }}\right) ^{\frac{1+\beta }{\alpha -1-\beta }}, \end{aligned}$$
(29)
$$\begin{aligned} K_{1}^{\sharp }&=\left( \frac{1+r}{r}\frac{c\beta }{k}\left. Q_{1} ^{\sharp }\right. ^{\alpha }\right) ^{\frac{1}{1+\beta }}, \end{aligned}$$
(30)
$$\begin{aligned} B_{1}^{\sharp }&=Q_{1}^{\sharp }-\left( {\overline{U}}-A_{1}\right) . \end{aligned}$$
(31)
Using the permit market equilibria,
$$\begin{aligned} {\underline{Q}}_{2}^{\sharp }&=2({\overline{U}}-A_{1})-{\underline{\Delta }} -Q_{1}^{\sharp }, \end{aligned}$$
(32)
$$\begin{aligned} {\overline{Q}}_{2}^{\sharp }&=2({\overline{U}}-A_{1})+{\overline{\Delta }} -Q_{1}^{\sharp }, \end{aligned}$$
(33)
Equation (29) provides an implicit expression for \(Q_{1}^{\sharp }\). In addition, note the following results:
$$\begin{aligned} \frac{dB_{1}^{\sharp }}{dr}<0,\text { }\frac{dB_{1}^{\sharp }}{dq}<0,\text { }\frac{dB_{1}^{\sharp }}{d{\underline{\Delta }}}<0,\text { }\frac{dB_{1}^{\sharp } }{d{\overline{\Delta }}}>0. \end{aligned}$$
Consistent with the intuition and the deterministic benchmark, banking decreases with the discount rate, the probability of a high cap and the size of this cap. On the opposite, a smaller low cap increases banking.

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 \(\beta \).

First order conditions and the permit market equilibrium conditions lead to the following optimal relationships between marginal cost functions at period 1 and at period 2 under certainty and under uncertainty (see “Deterministic Case Without Binding Irreversibility” and “Stochastic Case Without Binding Irreversibility” sections of Appendix 2):
$$\begin{aligned} C_{Q}(Q_{1}^{*})&=\frac{1}{1+r}C_{Q}(Q_{2}^{*})=\frac{1}{1+r}C_{Q} \left( \Gamma (q)-Q_{1}^{*}\right) ,\\ C_{Q}(Q_{1}^{\sharp })&=\frac{1}{1+r}\left[ qC_{Q}\left( {\underline{Q}} _{2}^{\sharp }\right) +(1-q)C_{Q}\left( {\overline{Q}}_{2}^{\sharp }\right) \right] \\&=\frac{1}{1+r}\left[ qC_{Q}\left( 2({\overline{U}}-A_{1})-{\underline{\Delta }} -Q_{1}^{\sharp }\right) +(1-q)C_{Q}\left( 2({\overline{U}}-A_{1})+{\overline{\Delta }} -Q_{1}^{\sharp }\right) \right] . \end{aligned}$$
If the marginal abatement cost is convex, a Jensen effect appears and this last optimality condition implies:
$$\begin{aligned} C_{Q}\left( Q_{1}^{\sharp }\right) >\frac{1}{1+r}C_{Q}\left( \Gamma (q)-Q_{1}^{\sharp }\right) . \end{aligned}$$
Therefore4 \(Q_{1}^{\sharp } >Q_{1}^{*}.\) The opposite holds if the marginal abatement cost is concave.
In the absence of banking, abatement at period 1 is: \(Q_{1}^{\text {wb} }={\overline{U}}-A_{1}\), as in the deterministic case. Investment takes place at both periods to decrease the abatement cost, up to the point where the marginal benefit of investment equals its marginal cost, and we get (see “Stochastic Case Without Binding Irreversibility” section of Appendix 2):
$$\begin{aligned} K_{1}^{\sharp \text {wb}}=\left[ \frac{1+r}{r}\frac{c\beta }{k}\left. Q_{1}^{\text {wb}}\right. ^{\alpha }\right] ^{\frac{1}{1+\beta }}. \end{aligned}$$
(34)
Therefore \(K_{1}^{\sharp \text {wb}}=K_{1}^{*\text {wb}}\). Without banking, uncertainty does not affect first period abatement and investment. Allowing positive banking \(Q_{1}^{\sharp }\ge Q_{1}^{\text {wb}}\), we have \(K_{1}^{\sharp }\ge K_{1}^{\sharp \text {wb}}\). In the absence of irreversibility, positive banking and investment in clean capital are complements, as it is the case in the benchmark without uncertainty. Banking allows firms to take advantage of uncertainty.

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

When the second period cap is uncertain and irreversibility is absent or non-binding:
  1. 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. 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

Program in period 1 is the same as that given by Eqs. (25) and (26). If the cap is increased in period 2 (lower-bar notation denotes this case), firm i abates \({\underline{Q}}_{2i}^{\sharp \text {ira}}\), purchases permits at a price \(\underline{p}_{2}\) but does not invest. On the contrary, if environmental policy happens to be more stringent (upper-bar notation denotes this case), firm i will abate \({\overline{Q}}_{2i}^{\sharp \text {ira} },\) purchase permits at a price \({\overline{p}}_{2}\) and invest \((\overline{K}_{2i}^{\sharp \text {ira}}-K_{1i}^{\sharp \text {ira}})\) in period 2. Firm \(i^{\prime }s\) programs for the second period then reads:
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}+{\underline{\Delta }}_{i}) =&\min _{{\underline{Q}}_{2i} }\left\{ C_{i}({\underline{Q}}_{2i},K_{1i})-{\underline{p}}_{2}\left[ B_{1i}+(A_{1i}+{\underline{\Delta }}_{i})-({\overline{U}}_{i}-{\underline{Q}} _{2i})\right] \right\} , \end{aligned}$$
(35)
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}-{\overline{\Delta }}_{i}) =&\min _{{\overline{Q}} _{2i},{\overline{K}}_{2i}}\left\{ C_{i}({\overline{Q}}_{2i},K_{1i})+k(\overline{K}_{2i}-K_{1i})\right. \nonumber \\&\left. -\,{\overline{p}}_{2}\left[ B_{1i}+(A_{1i}-{\overline{\Delta }} _{i})-({\overline{U}}_{i}-{\overline{Q}}_{2i})\right] \right\} . \end{aligned}$$
(36)
FOCs are listed in Table 2. Resolutions of the first and second period programs are given in “Stochastic Case with Binding Irreversibility” section of Appendix 2. At the aggregate level, we obtain:
$$\begin{aligned} (1+r)\left. Q_{1}^{\sharp \text {ira}}\right. ^{\alpha -1}= & {} q\left. {\underline{Q}}_{2}^{\sharp \text {ira}}\right. ^{\alpha -1}\nonumber \\&+\,(1-q)\left. {\overline{Q}}_{2}^{\sharp \text {ira}}\right. ^{\frac{\alpha -1-\beta }{1+\beta } }\left[ \frac{1+r}{r+q}\left( \left. Q_{1}^{\sharp \text {ira}}\right. ^{\alpha }+\frac{q}{1+r}\left. {\underline{Q}}_{2}^{\sharp \text {ira}}\right. ^{\alpha }\right) \right] ^{\frac{\beta }{1+\beta }}. \qquad \qquad \end{aligned}$$
(37)
Using the permit market equilibria:
$$\begin{aligned} {\underline{Q}}_{2}^{\sharp \text {ira}}&=\Gamma (q)-(1-q)(\underline{\Delta }+{\overline{\Delta }})-Q_{1}^{\sharp \text {ira}}, \end{aligned}$$
(38)
$$\begin{aligned} {\overline{Q}}_{2}^{\sharp \text {ira}}&=\Gamma (q)+q(\underline{\Delta }+{\overline{\Delta }})-Q_{1}^{\sharp \text {ira}}, \end{aligned}$$
(39)
Equation (37) provides an implicit expression for \(Q_{1} ^{\sharp \text {ira}}\).

4.2.2 Irreversibility Binding Regardless of the Second Period Cap

Program in period 1 is still given by Eqs. (25) and (26). Regardless of the observed cap, firms do not invest in period 2. If the cap is increased in period 2, firm i abates \({\underline{Q}}_{2i}^{\sharp \text {irb}} \), and purchases permits at a price \(\underline{p}_{2}\). On the contrary if environmental policy happens to be more stringent, firm i will abate \({\overline{Q}}_{2i}^{\sharp \text {irb}},\) and purchase permits at a price \({\overline{p}}_{2}\) in period 2. Firm \(i^{\prime }s\) programs for the second period then read:
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}+{\underline{\Delta }}_{i})&=\min _{{\underline{Q}}_{2i} }\left\{ C_{i}({\underline{Q}}_{2i},K_{1i})-{\underline{p}}_{2}\left[ B_{1i}+(A_{1i}+{\underline{\Delta }}_{i})-({\overline{U}}_{i}-{\underline{Q}} _{2i})\right] \right\} , \end{aligned}$$
(40)
$$\begin{aligned} V_{2i}(K_{1i},,A_{1i}-{\overline{\Delta }}_{i})&=\min _{{\overline{Q}}_{2i} }\left\{ C_{i}({\overline{Q}}_{2i},{\overline{K}}_{2i})-{\overline{p}}_{2}\left[ B_{1i}+(A_{1i}-{\overline{\Delta }}_{i})-({\overline{U}}_{i}-{\overline{Q}} _{2i})\right] \right\} . \end{aligned}$$
(41)
Again, FOCs are listed in Table 2 and resolutions of the first and second period programs are given in “Stochastic Case with Binding Irreversibility” section of Appendix 2. At the aggregate level, we obtain:
$$\begin{aligned} (1+r)^{\frac{1}{\alpha -1}}Q_{1}^{\sharp \text {irb}}=\left( q\left. {\underline{Q}}_{2}^{\sharp \text {irb}}\right. ^{\alpha -1}+(1-q)\left. {\overline{Q}}_{2}^{\sharp \text {irb}}\right. ^{\alpha -1}\right) ^{\frac{1}{\alpha -1}}. \end{aligned}$$
(42)
Using the permit market equilibria:
$$\begin{aligned} {\underline{Q}}_{2}^{\sharp \text {irb}}&=\Gamma (q)-(1-q)(\underline{\Delta }+{\overline{\Delta }})-Q_{1}^{\sharp \text {irb}}, \end{aligned}$$
(43)
$$\begin{aligned} {\overline{Q}}_{2}^{\sharp \text {irb}}&=\Gamma (q)+q(\underline{\Delta }+{\overline{\Delta }})-Q_{1}^{\sharp \text {irb}}, \end{aligned}$$
(44)
Equation (42) provides an implicit expression for \(Q_{1} ^{\sharp \text {irb}}\).
Comparing the abatement levels in the deterministic and stochastic cases when irreversibility is binding whatever the cap at period 2 yields:5 \(Q_{1}^{\sharp \text {irb}}>Q_{1}^{*\text {irr}}\) if and only if \(\alpha >2\). The argument runs as follows. As in the case without irreversibility, first order conditions lead to the following optimal relationships between marginal cost functions at period 1 and at period 2 without and with uncertainty:
$$\begin{aligned} C_{Q}(Q_{1}^{*\text {irr}})= & {} \frac{1}{1+r}C_{Q}(Q_{2}^{*\text {irr}}), \end{aligned}$$
(45)
$$\begin{aligned} C_{Q}(Q_{1}^{\sharp \text {irb}})= & {} \frac{1}{1+r}\left( qC_{Q}({\underline{Q}} _{2}^{\sharp \text {irb}})+(1-q)C_{Q}({\overline{Q}}_{2}^{\sharp \text {irb} })\right) . \end{aligned}$$
(46)
Since irreversibility is binding, the same amount of capital appears in the marginal abatement accross periods and does not therefore affect the optimal abatement (it cancels out in Eqs. (45) and (46)). Therefore the convexity of this “simplified marginal cost curve” only depends on parameter \(\alpha \): \(C_{Q}(Q_{1},K_{1})K_{1}^{\beta }=c\alpha Q_{1} ^{\alpha -1}.\) If the marginal abatement cost is convex (\(\alpha -1>1\Leftrightarrow \alpha >2\)), a Jensen effect appears and using the permits’ market equilibrium conditions, these optimality conditions imply:
$$\begin{aligned} C_{Q}(Q_{1}^{*\text {irr}})K_{1}^{*\text {irr}\beta }&=\frac{1}{1+r}C_{Q}(\Gamma (q)-Q_{1}^{*\text {irr}})K_{1}^{*\text {irr}\beta }\text {,}\\ C_{Q}(Q_{1}^{\sharp \text {irb}})K_{1}^{*\text {irb}\beta }&>\frac{1}{1+r}C_{Q}(\Gamma (q)-Q_{1}^{\sharp \text {irb}})K_{1}^{*\text {irb}\beta }, \end{aligned}$$
and therefore6 \(Q_{1}^{\sharp \text {irb}}>Q_{1}^{*\text {irr}}.\)
Fig. 1

Irreversibility frontiers and relevant first period abatement—dotted line \(r^{\sharp a}(q)\), dashed line \(r^{\sharp b}(q)\)

4.2.3 Irreversibility Frontiers

Now we can characterize the two irreversibility frontiers (see Fig. 1).
  • 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.

We have obtained a clear ranking of first period abatements analytically in the benchmark case and in the case with uncertainty when irreversibility is binding regardless of the cap: \(Q_{1}^{\sharp \text {irb}}>Q_{1}^{*\text {irr}}\) for \(\alpha >2,\) \(Q_{1}^{\sharp \text {irb}}\le Q_{1} ^{*\text {irr}}\) for \(\alpha \le 2\). In contrast, the ranking is ambiguous when irreversibility is binding only when the cap happens to be increased. However, it is easy to numerically find values for q and r such that \(Q_{1}^{\sharp \text {ira}}>Q_{1}^{*\text {irr}}\) and \(K_{1}^{\sharp \text {ira}}>K_{1}^{*\text {irr}}.\) There also exist combinations of q and r ensuring \(K_{1}^{\sharp \text {irb}}>K_{1}^{*\text {irr}}.\)

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. 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. 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 hold-up problem since firms would foresee the relationship between the first period investment and second period cap.

Footnotes

  1. 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. 2.

    Not necessarily in an optimal way. The cap can result from an international agreement.

  3. 3.

    That is, we do not specify how it is decided that firm i will get \(A_{1i}\) and \(A_{2i}\).

  4. 4.

    Recall that \(C_{QQ}(Q_{1})>0.\)

  5. 5.

    See “Stochastic Case with Binding Irreversibility” section of Appendix 2.

  6. 6.

    Recall that \(C_{QQ}(Q_{1})>0\)

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.City University of Hong KongKowloonHong Kong
  2. 2.Paris School of EconomicsUniversity Paris 1 Panthéon-SorbonneParisFrance
  3. 3.IREGE University of Savoie Mont BlancAnnecyFrance

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