### A.4. The case with $\gamma < 1$

We briefly examine the efficiency of free allowances under the alternative assumption $\gamma < 1$. First, without CBAM or under CBAM and reverse leakage, $\frac{d\mathcal{W}}{d\alpha}|_{\alpha=0} > 0$ in (18) if $\delta < \tau$ and $\gamma\delta > (1 - a^*(\bar{\theta}))\delta - \tau$. It implies that the welfare increases with $\alpha$ at zero and therefore $\alpha^* > 0$. Furthermore, substituting $\tau = \delta$ into (18) yields:

$$ \frac{d\mathcal{W}}{d\alpha} = q\delta[\gamma - \alpha]g(\bar{\theta})\frac{d\bar{\theta}}{d\alpha}. \qquad (27) $$

Since $\frac{d\mathcal{W}}{d\alpha} > 0$ if $\alpha < \gamma$ and $\frac{d\mathcal{W}}{d\alpha} < 0$ if $\alpha > \gamma$, which implies that $\mathcal{W}$ is increasing with $\alpha$ up to $\alpha = \gamma$ and decreasing with $\alpha$ for $\alpha > \gamma$. It is thus maximized at $\alpha^* = \gamma$.

Second, under CBAM and leakage, $\frac{d\mathcal{W}}{d\alpha}|_{\alpha=0} > 0$ in (19) if $\delta < \tau$ and $\gamma > 1 - a^*(\bar{\theta})$. Hence $\alpha^* > 0$ in this case. If $\tau = \delta$, $\frac{d\mathcal{W}}{d\alpha}$ is given by (27) so that $\alpha^* = 0$.

### A.5. Proof of Proposition 5

We consider in sequence the three cases described in Proposition 2.

(a) $p^{A\alpha} > c_f + \gamma\tau$: Leakage with and without CBAM.

The welfare without CBAM can be written as:

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{\alpha})}^{D(c_f)} [P(x) - c_f - \gamma\delta] dx - \delta\gamma D(c_f). \qquad (28) $$

The welfare with CBAM under leakage is:

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{\gamma\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{\gamma\alpha})}^{D(c_f + \gamma\tau)} [P(x) - c_f - \gamma\delta] dx - \delta\gamma D(c_f). \qquad (29) $$

Since $\bar{\theta}_{\gamma\alpha} > \bar{\theta}_{\alpha}$ and $D(c_f) > D(c_f + \gamma\tau)$, the welfare difference with CBAM minus without CBAM (29)–(28) writes:

$$ \Delta\mathcal{W} = \int_{\bar{\theta}_{\gamma\alpha}}^{\bar{\theta}_{\alpha}} \underbrace{[c_f + \gamma\delta - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta]}_{(i)} dG(\theta) \\ - \int_{D(c_f + \gamma\tau)}^{D(c_f)} \underbrace{[P(x) - c_f - \gamma\delta]}_{(ii)} dx. \qquad (30) $$

First, by (9) and because $\theta C(a^*(\theta)) + (1 - a^*(\theta))\tau$ is increasing with $\theta$, we have $c_f - c_h - \theta C(a^*(\theta)) > (1 - a^*(\theta) - \alpha)\tau$. The last inequality implies that (i) in (30) is higher than:

$$ \gamma\delta - \alpha\tau - (1 - a^*(\theta))[\delta - \tau], \qquad (31) $$

for every $\theta < \bar{\theta}_{\alpha}$. Since $\gamma \ge 1 \ge \alpha$, (31) is weakly higher than $[\gamma - (1 - a^*(\theta))] \ge 0$, where the last inequality is due to the fact that $\gamma \ge 1 \ge 1 - a^*(\theta)$ for every $\theta$ and $\tau \le \gamma$. Hence, (i) in (30) is strictly positive.

Second, for (ii) in (30), remark that $x > D(c_f + \gamma\tau)$ implies $P(x) < c_f + \gamma\tau$ by definition of $D(.) = P^{-1}(.)$ and $D'(.) < 0$. By $\tau \le \delta$, $P(x) < c_f + \gamma\tau$ implies $P(x) < c_f + \gamma\delta$ for every $x > D(c_f + \gamma\tau)$. Hence the second integral in the right-hand side of (30) is strictly negative.

We conclude $\Delta\mathcal{W} > 0$.

(b) $c_f + \gamma\tau > p^{A\alpha} > c_f$: leakage without CBAM and no leakage with CBAM.

The welfare without CBAM is given by (28), while the welfare with CBAM under no leakage is given by

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{A\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta). \qquad (32) $$

The welfare difference with and without CBAM (28) minus (32) is:

$$ \Delta\mathcal{W} = \int_{\bar{\theta}_{\gamma\alpha}}^{\bar{\theta}_{A\alpha}} [c_f + \gamma\delta - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{A\alpha})}^{D(c_f + \gamma\tau)} [P(x) - c_f - \gamma\delta] dx. $$

Proceeding as for leakage case (a) shows that $\Delta\mathcal{W} > 0$.
### A.4. The case with $\gamma < 1$

We briefly examine the efficiency of free allowances under the alternative assumption $\gamma < 1$. First, without CBAM or under CBAM and reverse leakage, $\frac{d\mathcal{W}}{d\alpha}|_{\alpha=0} > 0$ in (18) if $\delta < \tau$ and $\gamma\delta > (1 - a^*(\bar{\theta}))\delta - \tau$. It implies that the welfare increases with $\alpha$ at zero and therefore $\alpha^* > 0$. Furthermore, substituting $\tau = \delta$ into (18) yields:

$$ \frac{d\mathcal{W}}{d\alpha} = q\delta[\gamma - \alpha]g(\bar{\theta})\frac{d\bar{\theta}}{d\alpha}. \qquad (27) $$

Since $\frac{d\mathcal{W}}{d\alpha} > 0$ if $\alpha < \gamma$ and $\frac{d\mathcal{W}}{d\alpha} < 0$ if $\alpha > \gamma$, which implies that $\mathcal{W}$ is increasing with $\alpha$ up to $\alpha = \gamma$ and decreasing with $\alpha$ for $\alpha > \gamma$. It is thus maximized at $\alpha^* = \gamma$.

Second, under CBAM and leakage, $\frac{d\mathcal{W}}{d\alpha}|_{\alpha=0} > 0$ in (19) if $\delta < \tau$ and $\gamma > 1 - a^*(\bar{\theta})$. Hence $\alpha^* > 0$ in this case. If $\tau = \delta$, $\frac{d\mathcal{W}}{d\alpha}$ is given by (27) so that $\alpha^* = 0$.

### A.5. Proof of Proposition 5

We consider in sequence the three cases described in Proposition 2.

(a) $p^{A\alpha} > c_f + \gamma\tau$: Leakage with and without CBAM.

The welfare without CBAM can be written as:

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{\alpha})}^{D(c_f)} [P(x) - c_f - \gamma\delta] dx - \delta\gamma D(c_f). \qquad (28) $$

The welfare with CBAM under leakage is:

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{\gamma\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{\gamma\alpha})}^{D(c_f + \gamma\tau)} [P(x) - c_f - \gamma\delta] dx - \delta\gamma D(c_f). \qquad (29) $$

Since $\bar{\theta}_{\gamma\alpha} > \bar{\theta}_{\alpha}$ and $D(c_f) > D(c_f + \gamma\tau)$, the welfare difference with CBAM minus without CBAM (29)–(28) writes:

$$ \Delta\mathcal{W} = \int_{\bar{\theta}_{\gamma\alpha}}^{\bar{\theta}_{\alpha}} \underbrace{[c_f + \gamma\delta - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta]}_{(i)} dG(\theta) \\ - \int_{D(c_f + \gamma\tau)}^{D(c_f)} \underbrace{[P(x) - c_f - \gamma\delta]}_{(ii)} dx. \qquad (30) $$

First, by (9) and because $\theta C(a^*(\theta)) + (1 - a^*(\theta))\tau$ is increasing with $\theta$, we have $c_f - c_h - \theta C(a^*(\theta)) > (1 - a^*(\theta) - \alpha)\tau$. The last inequality implies that (i) in (30) is higher than:

$$ \gamma\delta - \alpha\tau - (1 - a^*(\theta))[\delta - \tau], \qquad (31) $$

for every $\theta < \bar{\theta}_{\alpha}$. Since $\gamma \ge 1 \ge \alpha$, (31) is weakly higher than $[\gamma - (1 - a^*(\theta))] \ge 0$, where the last inequality is due to the fact that $\gamma \ge 1 \ge 1 - a^*(\theta)$ for every $\theta$ and $\tau \le \gamma$. Hence, (i) in (30) is strictly positive.

Second, for (ii) in (30), remark that $x > D(c_f + \gamma\tau)$ implies $P(x) < c_f + \gamma\tau$ by definition of $D(.) = P^{-1}(.)$ and $D'(.) < 0$. By $\tau \le \delta$, $P(x) < c_f + \gamma\tau$ implies $P(x) < c_f + \gamma\delta$ for every $x > D(c_f + \gamma\tau)$. Hence the second integral in the right-hand side of (30) is strictly negative.

We conclude $\Delta\mathcal{W} > 0$.

(b) $c_f + \gamma\tau > p^{A\alpha} > c_f$: leakage without CBAM and no leakage with CBAM.

The welfare without CBAM is given by (28), while the welfare with CBAM under no leakage is given by

$$ \mathcal{W} = \int_{0}^{\bar{\theta}_{A\alpha}} [P(qG(\theta)) - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta). \qquad (32) $$

The welfare difference with and without CBAM (28) minus (32) is:

$$ \Delta\mathcal{W} = \int_{\bar{\theta}_{\gamma\alpha}}^{\bar{\theta}_{A\alpha}} [c_f + \gamma\delta - c_h - \theta C(a^*(\theta)) - (1 - a^*(\theta))\delta] dG(\theta) \\ + \int_{qG(\bar{\theta}_{A\alpha})}^{D(c_f + \gamma\tau)} [P(x) - c_f - \gamma\delta] dx. $$

Proceeding as for leakage case (a) shows that $\Delta\mathcal{W} > 0$.
(c) $c_f > p^{A\alpha}$: reverse leakage with and without CBAM.

The welfare is the same with or without CBAM for any given share of free allowances $\alpha$ because the equilibrium outcomes are the same. The CBAM is ineffective because no good is imported and the domestic price is the foreign price.

As for export rebates, they are effective only is case (c) of Proposition 3, in which case the welfare $W$ is defined in (14). Differentiating (14) with respect to $\alpha$, and using (4) and (9), yields (18). In A.3 we show that $\frac{dW}{d\alpha} > 0$ for every $\alpha < 1$. Hence $W$ increases with export rebates $\alpha > 0$.

### A.6. Impact of the carbon price

We now investigate the impact of the carbon price on the welfare with carbon leakage, accounting for territorial emissions first, and for the home country's contribution to total emissions next. Differentiating (15) with $E_T$ instead of $E_W$ with respect to $\tau$ and using (2) and (5) yields:

$$ \frac{dW}{d\tau} = \underbrace{[qG(\tilde{\theta}) - D(p_h)] \frac{dp_h}{d\tau}}_{\text{Price effect}} + q \underbrace{\int_{\theta}^{\tilde{\theta}} [\delta - \tau] \frac{da^*(\theta)}{d\tau} dG(\theta)}_{\text{Abatement effect}} + q[\tau(1 - a^*(\tilde{\theta}) - \alpha) - \delta(1 - a^*(\tilde{\theta}))]g(\tilde{\theta}) \frac{d\tilde{\theta}}{d\tau}. \quad (33) $$
$$ \underbrace{\text{Entry or exit effect}} $$

A marginal increase of the carbon price has three impacts on the welfare. First, a higher carbon price might increase the price of the good $p_h$ (in cases (a) and (b) but not (c)) which impacts positively firm's revenue but negatively consumer's spending. We call this channel the price effect. It corresponds to the right-hand term in the first line in (33). The price effect is negative if production $qG(\tilde{\theta})$ is lower than consumption $D(p_h)$, that is with imports (case (a)). It is nil under autarky (case (b)) since then $qG(\tilde{\theta}) = D(p_h)$: the increase of the good price is just a transfer from consumers to producers. With exports (case (c)), since $p_h = c_f$ (the domestic price is determined by the international price of the good), $\frac{dp_h}{d\tau} = 0$ so there is no price effect.

Second, pollution abatement improves the welfare by increasing pollution abatement. This abatement effect shows up the second line of (33). A marginal tax raise increases firm $\theta$'s abatement $a^*(\theta)$ by $\frac{da^*(\theta)}{d\tau} = \frac{1}{\theta C''(a^*(\theta))} > 0$, which reduces climate cost by $\delta$ while at the same time increases abatement cost by $\tau = \theta C'(a^*(\theta))$, where the last equality is due to (2). The abatement effect is nil with Pigou pricing $\tau = \delta$, and positive when carbon is under-priced $\tau < \delta$.

Third, a tax increase varies supply through entry or exit in the home country. We call this impact captured in the last line of (33) the entry or exit effect. As mentioned before, a higher tax favors entry if the threshold firm $\tilde{\theta}$ is a net seller of allowances (in which case $\frac{d\tilde{\theta}}{d\tau} > 0$), or induces some exists if it is a net buyer (then $\frac{d\tilde{\theta}}{d\tau} < 0$). The term into brackets in the third line of (33) is the difference between firm $\tilde{\theta}$'s regulatory cost $\tau(1 - a^*(\tilde{\theta})) - \tau\alpha$ and the climate cost $\delta(1 - a^*(\tilde{\theta}))$ per output. If the two coincide, e.g. under Pigou pricing $\tau = \delta$ and no free allowances $\alpha = 0$, the entry and exit effect is nil because firms internalize correctly the climate costs. Otherwise, the sign of the entry or exit effect depends upon both the difference between the regulatory and climate cost of firm $\tilde{\theta}$'s production, and firm $\tilde{\theta}$'s net position of in the allowance market (buyer or seller). If the regulation cost is too low - because carbon is under-priced $\tau < \delta$ and/or some allowances are free $\alpha > 1$ - then the entry and exit effect is negative when a higher carbon price favors entry, which turns out to be the case if the threshold firm is a net seller of allowances (i.e. if $\alpha > 1 - a^*(\tilde{\theta}))$. In contrast, it is positive when a higher carbon price make firms exit the industry, that is if the threshold firm is a net buyer of allowances (i.e. if $\alpha < 1 - a^*(\tilde{\theta}))$.

With global emissions $E_W$ in the welfare, differentiating (15) with respect to $\tau$ and using (2) and (5) yields:

$$ \frac{dW}{d\tau} = \underbrace{[qG(\tilde{\theta}) - D(p_h) - \delta\gamma D'(p_h)] \frac{dp_h}{d\tau}}_{\text{Price effect}} + q \underbrace{\int_{\theta}^{\tilde{\theta}} [\delta - \tau] \frac{da^*(\theta)}{d\tau} dG(\theta)}_{\text{Abatement effect}} + q[\tau(1 - a^*(\tilde{\theta}) - \alpha) - \delta(1 - a^*(\tilde{\theta}) - \gamma)]g(\tilde{\theta}) \frac{d\tilde{\theta}}{d\tau}. \quad (34) $$
$$ \underbrace{\text{Entry or exit effect}} $$

Compared to the case with territorial emissions in (33), the above relationship differs in two ways. First, the price effect takes into account the social gain of reduced emissions from lower consumption in the home country, i.e. the last term into brackets of the right-and term in the first line of (34). An increase of $p_h$ with a marginally higher $\tau$ decreases demand by $-D'(p_h)$ in cases (a) (with
(c) $c_f > p^{A\alpha}$: reverse leakage with and without CBAM.

The welfare is the same with or without CBAM for any given share of free allowances $\alpha$ because the equilibrium outcomes are the same. The CBAM is ineffective because no good is imported and the domestic price is the foreign price.

As for export rebates, they are effective only is case (c) of Proposition 3, in which case the welfare $W$ is defined in (14). Differentiating (14) with respect to $\alpha$, and using (4) and (9), yields (18). In A.3 we show that $\frac{dW}{d\alpha} > 0$ for every $\alpha < 1$. Hence $W$ increases with export rebates $\alpha > 0$.

### A.6. Impact of the carbon price

We now investigate the impact of the carbon price on the welfare with carbon leakage, accounting for territorial emissions first, and for the home country's contribution to total emissions next. Differentiating (15) with $E_T$ instead of $E_W$ with respect to $\tau$ and using (2) and (5) yields:

$$ \frac{dW}{d\tau} = \underbrace{[qG(\tilde{\theta}) - D(p_h)] \frac{dp_h}{d\tau}}_{\text{Price effect}} + q \underbrace{\int_{\theta}^{\tilde{\theta}} [\delta - \tau] \frac{da^*(\theta)}{d\tau} dG(\theta)}_{\text{Abatement effect}} + q[\tau(1 - a^*(\tilde{\theta}) - \alpha) - \delta(1 - a^*(\tilde{\theta}))]g(\tilde{\theta}) \frac{d\tilde{\theta}}{d\tau}. \quad (33) $$
$$ \underbrace{\text{Entry or exit effect}} $$

A marginal increase of the carbon price has three impacts on the welfare. First, a higher carbon price might increase the price of the good $p_h$ (in cases (a) and (b) but not (c)) which impacts positively firm's revenue but negatively consumer's spending. We call this channel the price effect. It corresponds to the right-hand term in the first line in (33). The price effect is negative if production $qG(\tilde{\theta})$ is lower than consumption $D(p_h)$, that is with imports (case (a)). It is nil under autarky (case (b)) since then $qG(\tilde{\theta}) = D(p_h)$: the increase of the good price is just a transfer from consumers to producers. With exports (case (c)), since $p_h = c_f$ (the domestic price is determined by the international price of the good), $\frac{dp_h}{d\tau} = 0$ so there is no price effect.

Second, pollution abatement improves the welfare by increasing pollution abatement. This abatement effect shows up the second line of (33). A marginal tax raise increases firm $\theta$'s abatement $a^*(\theta)$ by $\frac{da^*(\theta)}{d\tau} = \frac{1}{\theta C''(a^*(\theta))} > 0$, which reduces climate cost by $\delta$ while at the same time increases abatement cost by $\tau = \theta C'(a^*(\theta))$, where the last equality is due to (2). The abatement effect is nil with Pigou pricing $\tau = \delta$, and positive when carbon is under-priced $\tau < \delta$.

Third, a tax increase varies supply through entry or exit in the home country. We call this impact captured in the last line of (33) the entry or exit effect. As mentioned before, a higher tax favors entry if the threshold firm $\tilde{\theta}$ is a net seller of allowances (in which case $\frac{d\tilde{\theta}}{d\tau} > 0$), or induces some exists if it is a net buyer (then $\frac{d\tilde{\theta}}{d\tau} < 0$). The term into brackets in the third line of (33) is the difference between firm $\tilde{\theta}$'s regulatory cost $\tau(1 - a^*(\tilde{\theta})) - \tau\alpha$ and the climate cost $\delta(1 - a^*(\tilde{\theta}))$ per output. If the two coincide, e.g. under Pigou pricing $\tau = \delta$ and no free allowances $\alpha = 0$, the entry and exit effect is nil because firms internalize correctly the climate costs. Otherwise, the sign of the entry or exit effect depends upon both the difference between the regulatory and climate cost of firm $\tilde{\theta}$'s production, and firm $\tilde{\theta}$'s net position of in the allowance market (buyer or seller). If the regulation cost is too low - because carbon is under-priced $\tau < \delta$ and/or some allowances are free $\alpha > 1$ - then the entry and exit effect is negative when a higher carbon price favors entry, which turns out to be the case if the threshold firm is a net seller of allowances (i.e. if $\alpha > 1 - a^*(\tilde{\theta}))$. In contrast, it is positive when a higher carbon price make firms exit the industry, that is if the threshold firm is a net buyer of allowances (i.e. if $\alpha < 1 - a^*(\tilde{\theta}))$.

With global emissions $E_W$ in the welfare, differentiating (15) with respect to $\tau$ and using (2) and (5) yields:

$$ \frac{dW}{d\tau} = \underbrace{[qG(\tilde{\theta}) - D(p_h) - \delta\gamma D'(p_h)] \frac{dp_h}{d\tau}}_{\text{Price effect}} + q \underbrace{\int_{\theta}^{\tilde{\theta}} [\delta - \tau] \frac{da^*(\theta)}{d\tau} dG(\theta)}_{\text{Abatement effect}} + q[\tau(1 - a^*(\tilde{\theta}) - \alpha) - \delta(1 - a^*(\tilde{\theta}) - \gamma)]g(\tilde{\theta}) \frac{d\tilde{\theta}}{d\tau}. \quad (34) $$
$$ \underbrace{\text{Entry or exit effect}} $$

Compared to the case with territorial emissions in (33), the above relationship differs in two ways. First, the price effect takes into account the social gain of reduced emissions from lower consumption in the home country, i.e. the last term into brackets of the right-and term in the first line of (34). An increase of $p_h$ with a marginally higher $\tau$ decreases demand by $-D'(p_h)$ in cases (a) (with
<figure><img src="image_11.png" /><figcaption>Fig. 9. Equilibrium prices with CBAM.</figcaption></figure>

imports), which reduces emissions by $γ$ and has social value $δ$. The marginal climate gain from the price increase with imports is therefore $-\delta\gamma D'(p_h) > 0$. Second, the entry or exit effect measures the carbon impact of the threshold firm's output relative to the foreign alternative rather in absolute term, i.e. with $1 - a^*(\theta) - \gamma$ rather than $1 - a^*(\theta)$. It therefore lower and can even be positive if $1 - a^*(\theta) < \gamma$, in which case firm $\tilde{\theta}$'s production improves the welfare by replacing more carbon-intensive foreign products.

### A.7. Additional figures

See Figs. 9–11.

### A.8. Robustness of the quantitative analysis

In this section, we gauge the robustness of our quantitative results with respect to the calibrated parameters. First, we estimate $\beta$ using different years. Using the Italian plant-level data for years other than 2019, we find $\hat{\beta} = 1.48$ for 2018 (using the average $\tau$ of 15 observed in that year), and $\hat{\beta} = 0.77$ for 2017 (using the average $\tau$ of 5).<fnref n="35" /> In Fig. 12 we plot the welfare under CBAM in each sector (as in Fig. 7) using the $\beta$ estimated in different years.

The graph shows that the welfare is close to the baseline welfare for any level for $\alpha$, but is increasing in the convexity parameter $\beta$. Intuitively, when the cost function is more convex, the abatement costs are on average higher, thus the welfare gain arising from the CBAM “protecting” domestic producers from foreign competition becomes larger.

Second, we evaluate the robustness of our results with respect to the values for domestic and foreign emissions. We set $\gamma = 1$, which means that the foreign emission factors are equal to the domestic ones, before abatement.

In Fig. 13, we can see that the welfare under CBAM is essentially the same as in the baseline for cement, while it is a bit higher for the steel sector. This is because in the steel sector global emissions are significantly lower than in the baseline, as $\gamma$ goes from 1.42 to 1. Overall, these robustness exercises indicate that our welfare results are not driven, neither qualitatively nor quantitatively, by the specific point estimates that we impose in our baseline calibration.

<footnote n="35">³⁵ If instead we use the same $\tau$ as in 2019, we find $\hat{\beta} = 1.67$ in 2018 and $\hat{\beta} = 1.09$ in 2017.</footnote>
<figure><img src="image_11.png" /><figcaption>Fig. 9. Equilibrium prices with CBAM.</figcaption></figure>

imports), which reduces emissions by $γ$ and has social value $δ$. The marginal climate gain from the price increase with imports is therefore $-\delta\gamma D'(p_h) > 0$. Second, the entry or exit effect measures the carbon impact of the threshold firm's output relative to the foreign alternative rather in absolute term, i.e. with $1 - a^*(\theta) - \gamma$ rather than $1 - a^*(\theta)$. It therefore lower and can even be positive if $1 - a^*(\theta) < \gamma$, in which case firm $\tilde{\theta}$'s production improves the welfare by replacing more carbon-intensive foreign products.

## A.7. Additional figures

See Figs. 9–11.

## A.8. Robustness of the quantitative analysis

In this section, we gauge the robustness of our quantitative results with respect to the calibrated parameters. First, we estimate $\beta$ using different years. Using the Italian plant-level data for years other than 2019, we find $\hat{\beta} = 1.48$ for 2018 (using the average $\tau$ of 15 observed in that year), and $\hat{\beta} = 0.77$ for 2017 (using the average $\tau$ of 5).<fnref n="35" /> In Fig. 12 we plot the welfare under CBAM in each sector (as in Fig. 7) using the $\beta$ estimated in different years.

The graph shows that the welfare is close to the baseline welfare for any level for $\alpha$, but is increasing in the convexity parameter $\beta$. Intuitively, when the cost function is more convex, the abatement costs are on average higher, thus the welfare gain arising from the CBAM “protecting” domestic producers from foreign competition becomes larger.

Second, we evaluate the robustness of our results with respect to the values for domestic and foreign emissions. We set $\gamma = 1$, which means that the foreign emission factors are equal to the domestic ones, before abatement.

In Fig. 13, we can see that the welfare under CBAM is essentially the same as in the baseline for cement, while it is a bit higher for the steel sector. This is because in the steel sector global emissions are significantly lower than in the baseline, as $\gamma$ goes from 1.42 to 1. Overall, these robustness exercises indicate that our welfare results are not driven, neither qualitatively nor quantitatively, by the specific point estimates that we impose in our baseline calibration.

<footnote n="35">³⁵ If instead we use the same $\tau$ as in 2019, we find $\hat{\beta} = 1.67$ in 2018 and $\hat{\beta} = 1.09$ in 2017.</footnote>
<figure><img src="image_12.png" /><figcaption>Fig. 10. Cutoff emission intensity, $r = 162$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Emissions without CBAM, with $r = 162$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10. Cutoff emission intensity, $r = 162$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Emissions without CBAM, with $r = 162$.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 12. Welfare with CBAM with $\beta$ calibrated in different years.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Welfare with CBAM with $\gamma = 1$.</figcaption></figure>

References

Adão, Rodrigo, Arkolakis, Costas, Esposito, Federico, 2019. General Equilibrium Effects in Space: Theory and Measurement. Vol. 25544, National Bureau of Economic Research, Inc.

Ahlvik, Lassi, Liski, Matti, 2019. Think Global, Act Local! a Mechanism for Global Commons and Mobile Firms. CESifo Working Paper Series 7597, CESifo.
<figure><img src="image_14.png" /><figcaption>Fig. 12. Welfare with CBAM with $\beta$ calibrated in different years.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Welfare with CBAM with $\gamma = 1$.</figcaption></figure>

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Böhringer, C., Schneider, J., Asane-Otoo, E., 2021. Trade in carbon and carbon tariffs. Environ. Resour. Econ. 78 (4), 669-708.

Branger, Frederic, Quirion, Philippe, 2014. Would border carbon adjustments prevent carbon leakage and heavy industry competitiveness losses? Insights from a meta-analysis of recent economic studies. Ecol. Econom. 99 (C), 29-39.

Branger, Frédéric, Quirion, Philippe, Chevallier, Julien, 2016. Carbon leakage and competitiveness of cement and steel industries under the EU ETS: much ado about nothing. Energy J. 37 (3), 109-136.

Bushnell, James B., Chong, Howard, Mansur, Erin T., 2013. Profiting from regulation: Evidence from the European carbon market. Am. Econ. J.: Econ. Policy 5 (4), 78-106.

Caliendo, Lorenzo, Parro, Fernando, 2015. Estimates of the trade and welfare effects of NAFTA. Rev. Econom. Stud. 82 (1), 1-44.

Cembureau, 2019. 2019 Activity report.

Cicala, Steve, Hémous, David, Olsen, Morten, 2022. Adverse selection as a policy instrument: Unraveling climate change. In: NBER Working Paper.

Cosbey, Aaron, Droege, Susanne, Fischer, Carolyn, Munnings, Clayton, 2019. Developing guidance for implementing border carbon adjustments: Lessons, cautions, and research needs from the literature. Rev. Environ. Econ. Policy 13 (1), 3-22.

Dechezleprêtre, Antoine, Gennaioli, Caterina, Martin, Ralf, Muûls, Mirabelle, Stoerk, Thomas, 2022. Searching for carbon leaks in multinational companies. J. Environ. Econ. Manag. 112, 102601.

Esposito, Federico, 2020. Estimating the welfare costs of autarky: a sufficient statistics approach. Econom. Lett. 194, 109361.

European Commission, 2021a. Proposal for a regulation of the European parliament and of the council establishing a carbon border adjustment mechanism.

European Commission, 2021b. Towards competitive and clean European steel, commission staff working document.

European Court of Auditors, 2020. The eu's emissions trading system: free allocation of allowances needed better targeting. URL https://www.eca.europa.eu/Lists/ECADocuments/SR20_18/SR_EU-ETS_EN.pdf.

European Parliament, 2009. Directive 2009/29/EC of the European parliament and the council of 23 april 2009. Off. J. Eur. Union.

Farrokhi, Farid, Lashkaripour, Ahmad, 2022. Can trade policy mitigate climate change. In: CEPR Working paper.

Fischer, Carolyn, Fox, Alan K., 2012. Comparing policies to combat emissions leakage: Border carbon adjustments versus rebates. J. Environ. Econ. Manag. 64 (2), 199-216.

Fowlie, Meredith L., Reguant, Mar, 2022. Mitigating emissions leakage in incomplete carbon markets. J. Assoc. Environ. Resour. Econ. 9 (2), 307-343.

Fowlie, Meredith, Reguant, Mar, Ryan, Stephen P., 2016. Market-based emissions regulation and industry dynamics. J. Polit. Econ. 124 (1), 249-302.

Global Cement and Concrete Association, 2022. GNR project: Reporting CO₂.

Healy, Sean, Schumacher, Katja, Eichhammer, Wolfgang, 2018. Analysis of carbon leakage under phase III of the EU emissions trading system: Trading patterns in the cement and aluminium sectors. Energies 11 (5), 1231.

Hopenhayn, Hugo A., 1992. Entry, exit, and firm long run equilibria. Econometrica 60 (5), 1127-1150.

Kortum, Samuel, Weisbach, David A., 2021. Optimal Unilateral Carbon Policy. CESifo Working Paper Series 9409, CESifo.

Levinson, Arik, 2023. Are developed countries outsourcing pollution? J. Econ. Perspect. 37 (3), 87-110.

Maratou, Alexandra, 2021. EU carbon border adjustment mechanism and implications for Turkey. URL https://ercst.org/wp-content/uploads/2021/07/20211006_Turkey_CBAM_EU-Turkey-climate-policy-dialogue_v3.pdf.

Markusen, James, 1975. International externalities and optimal tax structures. J. Int. Econ. 5, 15-29.

Martin, Ralf, Muûls, Mirabelle, de Preux, Laure B., Wagner, Ulrich J., 2014. Industry compensation under relocation risk: A firm-level analysis of the EU emissions trading scheme. Amer. Econ. Rev. 104 (8), 2482-2508.

Melitz, Marc J., 2003. The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica 71 (6), 1695-1725.

Naegele, Helene, Zaklan, Aleksandar, 2019. Does the EU ETS cause carbon leakage in European manufacturing? J. Environ. Econ. Manag. 93, 125-147.

Reinaud, Julia, 2005. Industrial competitiveness under the European union emissions trading scheme. IEA information paper. Oil Gas Energy Law.

Rennert, Kevin, Errickson, Frank, Prest, Brian C, Rennels, Lisa, Newell, Richard G, Pizer, William, Kingdon, Cora, Wingenroth, Jordan, Cooke, Roger, Parthum, Bryan, et al., 2022. Comprehensive evidence implies a higher social cost of CO2. Nature 610 (7933), 687-692.

Saussay, Aurélien, Zugravu-Soilita, Natalia, 2023. International production chains and the pollution offshoring hypothesis: An empirical investigation. Resour. Energy Econ. 73, 101357.

Schmalensee, Richard, Stavins, Robert, 2017. Lessons learned from three decades of experience with cap and trade. Rev. Environ. Econ. Policy 11 (1), 59-79.

Wagner, Ulrich J., Martin, Ralf, Muûls, Mirabelle, de Preux, Laure B., 2014. The causal effects of the European union emissions trading scheme: Evidence from french manufacturing plants. IZA Conf. 1020.

Weisbach, David, Kortum, Samuel S., Wang, Michael, Yao, Yujia, 2023. Trade, Leakage, and the Design of a Carbon Tax. Tech. rep., NBER Working Paper.

World Steel Association, 2020. Sustainable steel: Indicators 2020 and steel applications. URL https://worldsteel.org/wp-content/uploads/Sustainable-steel.pdf.

Wörtler, Martin, Schuler, Felix, Voigt, Nicole, Schmidt, Torben, Dahlmann, Peter, Lüngen, Hans Bodo, Ghenda, Jean-Theo, 2013. Steel's Contribution to a Low-Carbon Europe 2050: Technical and Economic Analysis of the Steel Sector's CO2 Abatement Potential. Boston Consulting Group.
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Böhringer, Christoph, Fischer, Carolyn, Rosendahl, Knut Einar, 2014. Cost-effective unilateral climate policy design: Size matters. J. Environ. Econ. Manag. 67 (3), 318-339.

Böhringer, Christoph, Fischer, Carolyn, Rosendahl, Knut Einar, Rutherford, Thomas Fox, 2022. Potential impacts and challenges of border carbon adjustments. Nature Clim. Change 12 (1), 22-29.

Böhringer, C., Schneider, J., Asane-Otoo, E., 2021. Trade in carbon and carbon tariffs. Environ. Resour. Econ. 78 (4), 669-708.

Branger, Frederic, Quirion, Philippe, 2014. Would border carbon adjustments prevent carbon leakage and heavy industry competitiveness losses? Insights from a meta-analysis of recent economic studies. Ecol. Econom. 99 (C), 29-39.

Branger, Frédéric, Quirion, Philippe, Chevallier, Julien, 2016. Carbon leakage and competitiveness of cement and steel industries under the EU ETS: much ado about nothing. Energy J. 37 (3), 109-136.

Bushnell, James B., Chong, Howard, Mansur, Erin T., 2013. Profiting from regulation: Evidence from the European carbon market. Am. Econ. J.: Econ. Policy 5 (4), 78-106.

Caliendo, Lorenzo, Parro, Fernando, 2015. Estimates of the trade and welfare effects of NAFTA. Rev. Econom. Stud. 82 (1), 1-44.

Cembureau, 2019. 2019 Activity report.

Cicala, Steve, Hémous, David, Olsen, Morten, 2022. Adverse selection as a policy instrument: Unraveling climate change. In: NBER Working Paper.

Cosbey, Aaron, Droege, Susanne, Fischer, Carolyn, Munnings, Clayton, 2019. Developing guidance for implementing border carbon adjustments: Lessons, cautions, and research needs from the literature. Rev. Environ. Econ. Policy 13 (1), 3-22.

Dechezleprêtre, Antoine, Gennaioli, Caterina, Martin, Ralf, Muûls, Mirabelle, Stoerk, Thomas, 2022. Searching for carbon leaks in multinational companies. J. Environ. Econ. Manag. 112, 102601.

Esposito, Federico, 2020. Estimating the welfare costs of autarky: a sufficient statistics approach. Econom. Lett. 194, 109361.

European Commission, 2021a. Proposal for a regulation of the European parliament and of the council establishing a carbon border adjustment mechanism.

European Commission, 2021b. Towards competitive and clean European steel, commission staff working document.

European Court of Auditors, 2020. The eu's emissions trading system: free allocation of allowances needed better targeting. URL https://www.eca.europa.eu/Lists/ECADocuments/SR20_18/SR_EU-ETS_EN.pdf.

European Parliament, 2009. Directive 2009/29/EC of the European parliament and the council of 23 april 2009. Off. J. Eur. Union.

Farrokhi, Farid, Lashkaripour, Ahmad, 2022. Can trade policy mitigate climate change. In: CEPR Working paper.

Fischer, Carolyn, Fox, Alan K., 2012. Comparing policies to combat emissions leakage: Border carbon adjustments versus rebates. J. Environ. Econ. Manag. 64 (2), 199-216.

Fowlie, Meredith L., Reguant, Mar, 2022. Mitigating emissions leakage in incomplete carbon markets. J. Assoc. Environ. Resour. Econ. 9 (2), 307-343.

Fowlie, Meredith, Reguant, Mar, Ryan, Stephen P., 2016. Market-based emissions regulation and industry dynamics. J. Polit. Econ. 124 (1), 249-302.

Global Cement and Concrete Association, 2022. GNR project: Reporting CO₂.

Healy, Sean, Schumacher, Katja, Eichhammer, Wolfgang, 2018. Analysis of carbon leakage under phase III of the EU emissions trading system: Trading patterns in the cement and aluminium sectors. Energies 11 (5), 1231.

Hopenhayn, Hugo A., 1992. Entry, exit, and firm long run equilibria. Econometrica 60 (5), 1127-1150.

Kortum, Samuel, Weisbach, David A., 2021. Optimal Unilateral Carbon Policy. CESifo Working Paper Series 9409, CESifo.

Levinson, Arik, 2023. Are developed countries outsourcing pollution? J. Econ. Perspect. 37 (3), 87-110.

Maratou, Alexandra, 2021. EU carbon border adjustment mechanism and implications for Turkey. URL https://ercst.org/wp-content/uploads/2021/07/20211006_Turkey_CBAM_EU-Turkey-climate-policy-dialogue_v3.pdf.

Markusen, James, 1975. International externalities and optimal tax structures. J. Int. Econ. 5, 15-29.

Martin, Ralf, Muûls, Mirabelle, de Preux, Laure B., Wagner, Ulrich J., 2014. Industry compensation under relocation risk: A firm-level analysis of the EU emissions trading scheme. Amer. Econ. Rev. 104 (8), 2482-2508.

Melitz, Marc J., 2003. The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica 71 (6), 1695-1725.

Naegele, Helene, Zaklan, Aleksandar, 2019. Does the EU ETS cause carbon leakage in European manufacturing? J. Environ. Econ. Manag. 93, 125-147.

Reinaud, Julia, 2005. Industrial competitiveness under the European union emissions trading scheme. IEA information paper. Oil Gas Energy Law.

Rennert, Kevin, Errickson, Frank, Prest, Brian C, Rennels, Lisa, Newell, Richard G, Pizer, William, Kingdon, Cora, Wingenroth, Jordan, Cooke, Roger, Parthum, Bryan, et al., 2022. Comprehensive evidence implies a higher social cost of CO2. Nature 610 (7933), 687-692.

Saussay, Aurélien, Zugravu-Soilita, Natalia, 2023. International production chains and the pollution offshoring hypothesis: An empirical investigation. Resour. Energy Econ. 73, 101357.

Schmalensee, Richard, Stavins, Robert, 2017. Lessons learned from three decades of experience with cap and trade. Rev. Environ. Econ. Policy 11 (1), 59-79.

Wagner, Ulrich J., Martin, Ralf, Muûls, Mirabelle, de Preux, Laure B., 2014. The causal effects of the European union emissions trading scheme: Evidence from french manufacturing plants. IZA Conf. 1000.

Weisbach, David, Kortum, Samuel S., Wang, Michael, Yao, Yujia, 2023. Trade, Leakage, and the Design of a Carbon Tax. Tech. rep., NBER Working Paper.

World Steel Association, 2020. Sustainable steel: Indicators 2020 and steel applications. URL https://worldsteel.org/wp-content/uploads/Sustainable-steel.pdf.

Wörtler, Martin, Schuler, Felix, Voigt, Nicole, Schmidt, Torben, Dahlmann, Peter, Lüngen, Hans Bodo, Ghenda, Jean-Theo, 2013. Steel's Contribution to a Low-Carbon Europe 2050: Technical and Economic Analysis of the Steel Sector's CO2 Abatement Potential. Boston Consulting Group.