Rushing the Impatient: Allowance Reserves and the Time Profile of Low-Carbon Investments

Abstract

Postponing the issue date of allowances in a cap-and-trade scheme, by e.g. a reserve mechanism, impacts the time profile of low-carbon investments. If the postponement constrains intertemporal arbitrage, short-term investments increase but long-term investments are deterred. This effect aggravates the shortage of long-term investments at least partially attributed to firms’ impatience. The cancellation of allowances agreed for Phase IV of the EU ETS is suitable to counteract the negative effects of cap-neutral postponement on long-term investments—by making the reserve redundant. All effects crucially depend on how firms form expectations about future allowance prices.

Appendix

1.1 1.1 Proof of Conditions (21) and (22)

We use the difference between adoption and no adoption for each technology, i.e. $C(e^a_t,x^a_t)-C(e^a_{t,I},x^a_{t,I})$. A firm will only invest in a technology if and only if this difference is greater zero. Hence:

$$\begin{aligned} \begin{aligned}&\frac{c}{2}(u-e^*_1)^2+p^a_1 x^*_1+\frac{c}{2(1+r^p)}(u-e^*_2)^2+\frac{p^a_2 x^*_2}{1+r^p} \\&\quad -\frac{c}{2}(u-e^a_{1,I})^2+p^a_1 x^a_{1,I}+\frac{c}{2(1+r^p)}(u-e^a_{2,I})^2+\frac{p^a_2 x^a_{2,I}}{1+r^p} - f_t >0 \end{aligned} \end{aligned}$$

Substituting in $e^a_t$ and $e^a_{t,I}$ from conditions (5) and (20) leads to the cancellation of the abatement cost parts of the inequality, leaving:

$$\begin{aligned} p^a_1(x^*_1 - x^a_{1,I}) + \frac{p^a_2}{1+r}(x^*_2-x^a_{2,I}) > f_t. \end{aligned}$$

In the banking scenario, we use condition (6) to replace $p_2$ by $(1+r^p)p_1$. Additionally, it must hold that $x^a_1+x^a_2=e^a_1+e^a_2-b_0$. Using this, the inequality above simplifies to $p^a_1 u(1-\theta _t)>f_t$. In the no-banking scenario, we can determine purchases and sales for each period separately, i.e. $x^*_t=e^*_t - b_0$ and $x^a_{t,I} = e^a_{t,I}$. Plugging this into the inequality above yields $p_1 u(1-\theta _1)>f_1$ for technology 1 and $[p^a_2 u(1-\theta _2)]/(1+r^p) > f_2$ for technology 2. Thus, using aggregate values, it holds that a technology is adopted by a firm if and only if

$$\begin{aligned} \frac{p^a_t u}{(1+r^p)^{t-1}}(1-\theta _t) > f_t. \end{aligned}$$

1.2 1.2 Proof of Lemma 3

Conditions (23) and (24) directly follow from combining condition (6) with the reduced baseline emissions with technology adoption. The thresholds for $\bar{E}$ are calculated as follows: Solve (22) for $p^a_j$, plug into (20) and use the definition of total emissions with adoption [shown right after (20)] to compute thresholds for different diffusion scenarios (such as $\gamma ^a_1 = \gamma ^a_2 = 0$ or $\gamma ^a_1 = 1$ and $\gamma ^a_2 = 0$ etc.).

1.3 1.3 Proof of Lemma 4

Analogue to proof of Lemma 3 but using (16) and (17) instead of total emissions.

1.4 1.4 Derivation of $\hat{\hbox {R}}$

Lemma 3 defines diffusion shares under banking. Explicitly adding the amount of allowances whose issuance gets postponed to Condition 23, setting it equal to zero and solving for R yields the critical value $\hat{R}$ at which firms are indifferent between banking and not banking.

$$\begin{aligned} \begin{aligned} \hat{R}=\frac{u}{2+r^p}[(1+r^p)\gamma _1^{b,a}(1-\theta _1)- \gamma _2^{b,a}(1-\theta _2)]+\frac{(1+r)(B_0+S_1)-S_2-r^pu}{2+r^p} \end{aligned} \end{aligned}.$$

Hence, for $R \le \hat{R}$ there is no incentive to bank while for $R<\hat{R}$, there is an incentive to bank.

To elicit $\hat{R}$ with respect to a specific long-run cap, it is now necessary to replace the diffusion shares with the corresponding value from Lemma 3. In the case presented in Fig. 1, the long-run cap fulfills the second condition, i.e. $\gamma _1^{b,a}=\frac{2u - \bar{E}}{u (1 - \theta _1)} - \frac{2 + r^p}{c u^2(1-\theta _1)^2} f_1$ and $\gamma _2^{b,a}=0$. Plugging these values into the above equation for $\hat{R}$ yields

$$\begin{aligned} \begin{aligned} \hat{R}_{example}=u-S_2-\frac{(1+r)f_1}{cu(1-\theta _1)} \end{aligned} \end{aligned}$$