Between a Rock and a Hard Place: A Trade-Theory Analysis of Leakage Under Production- and Consumption-Based Policies

Abstract

Without a comprehensive global climate agreement, carbon leakage remains a contentious issue. Consumption-based pricing of emissions—which could in practice be implemented with a full border tax adjustment (BTA)—has been forwarded as an option to increase the effectiveness of unilateral climate policy. This paper questions the economic rationale behind this approach, using a theoretical $2 \times 2$ trade model in which leakage occurs through terms-of-trade effects. We show analytically, first, that consumption-based pricing of emissions does not necessarily result in less leakage than production-based policies. Second, the sign of the optimal unilateral carbon tariff depends on the carbon-intensity differential between the foreign country’s exporting and non-exporting sectors, and not on the differential between home’s and foreign’s exporting sectors, as implied by the full BTA approach. Third, based on empirical data for the year 2004, our model implies that full BTA applied by the European Union on e.g. imports from and exports to China would—by shifting China’s production from the export sector with a relatively low carbon-intensity towards the more carbon-intensive non-export sector—actually increase leakage.

Appendix

Table 1 Average carbon intensities of imports, exports, total production, non-export sector and measured relative to each other for the EU27 and G20 countries not part of the EU27 (excl. Saudi Arabia)

Proof of Lemma 2

The market clearing condition for the $Y$ world market reads: $M^{Yh} -E^{Yf} =0$, where both $M^{Yh}$ and $E^{Yf}$ depend upon the world market price $p^{*}$. Moreover, the budget balance condition Eq. (2) implies: $p^{*}M^{Yh} =E^{Xh}$. $E^{Xh}$ depends upon $p^{*}$ as well as $\mu$, which translates into $M^{Yh}$. This implies for the derivative of the market clearing condition with respect to a change in Home’s carbon tax $\mu$ (equivalent to a change of the cap $\bar{{Z}}^{h})$: $M^{Yh} (p^{*},\mu )-E^{Yf} (p^{*})=0$ and thus

$$\begin{aligned} \frac{\partial M^{Yh}}{\partial \mu }+\frac{\partial M^{Yh}}{\partial p^{*}}\frac{dp^{*}}{d\mu }-\frac{dE^{Yf}}{dp^{*}}\frac{dp^{*}}{d\mu }=0\quad \Rightarrow \quad \frac{dp^{*}}{d\mu }=\frac{-\frac{\partial M^{Yh}}{\partial \mu }}{\left( {\frac{\partial M^{Yh}}{\partial p^{*}}-\frac{dE^{Yf}}{dp^{*}}} \right) } \end{aligned}$$

(30)

Under standard conditions, the denominator is unambiguously negative (note that in the present setting both Home and Foreign behave as price-takers). Hence, we only need to compute the sign of the numerator. Using the fact that consumers’ preferences are homothetic, Home’s imports, $M^{Yh} = C^{Yh} -Q^{Yh}$, can be rewritten by inserting the standard expressions for the consumption of $Y$ derived from utility maximizing given an income budget

$$\begin{aligned} M^{Yh} = \frac{\bar{{\eta }}}{q} I^{h}-Q^{Yh} \end{aligned}$$

(31)

where $\eta (q)$ denotes the share of Home’s real income, $I^{h}$, spent on good $X$, and $\overline{{\eta }}(q)=1-\eta (q)$ denotes the share of Home’s real income spent on good $Y.$ Consumption of $Y$ decreases in the relative consumer price of $Y$, denoted by $q$. Since for the moment there are no consumer taxes in place, it is $q$ = $p^{*}$. Since both goods, $X$ and $Y$, are traded internationally, total income is expressed by applying the world market price $p^{*}$, i.e. as $Q^{Xh}+ p^{*}Q^{Yh}$, implying:

$$\begin{aligned} M^{Yh} = \frac{\overline{{\eta }}}{p^{*}}Q^{Xh} -\eta Q^{Yh} =\frac{\overline{{\eta }}}{p^{*}}T^{h}(Q^{Yh} )-\eta Q^{Yh} \end{aligned}$$

(32)

Since in Eq. (32) only $Q^{Yh}$ depends explicitly on the production tax, $\mu$, we find

$$\begin{aligned} \frac{\partial M^{Yh}}{\partial \mu }=\left( {\frac{\overline{{\eta }}}{p^{*}}T_{Q^{Yh}}^h -\eta } \right) \frac{\partial Q^{Yh} }{\partial \mu }=-\left( {\bar{{\eta }}\frac{p}{p^{*}}+\eta } \right) \frac{\partial Q^{Yh} }{\partial \mu } \end{aligned}$$

(33)

Differentiating the first and the second part of the producers’ efficiency condition in Eq. (19) with respect to $\mu$ and solving for $\frac{\partial Q^{Yh}}{\partial \mu }$ yields

$$\begin{aligned} \frac{\partial Q^{Yh}}{\partial \mu }=\frac{\gamma ^{Yh} -q\gamma ^{Xh} }{\left( {1-\gamma ^{Xh} \mu } \right) ^{2} T_{Q^{Yh}Q^{Yh}}^h } \end{aligned}$$

(34)

Confronting the last equation with Eq. (14) shows that it will be negative when $\gamma ^{Yh} >q \gamma ^{Xh}$. Therefore, Eq. (33) will be positive if and only if $\gamma ^{Yh} >p^{*} \gamma ^{Xh}$, where $p^{*}= q$. In this case, Eq. (30) will be positive as well.$\square$

Proof of Lemma 3

To determine the effect of $\sigma$ on $p^{*}$, we start by differentiating the market clearing condition, obtaining (cf. Eq. 30):

$$\begin{aligned} \frac{dp^{*}}{d\sigma } = -\frac{\frac{\partial M^{Yh} }{\partial \sigma }}{\frac{\partial M^{Yh} }{\partial p^{*}}-\frac{\partial E^{Yf} }{\partial p^{*}}} \end{aligned}$$

(35)

Again, the denominator is negative under standard conditions. To determine the sign of the numerator, we start by further specifying Home’s consumption. With homothetic preferences, we have:

$$\begin{aligned} \frac{ C^{Xh} }{C^{Yh} }=\frac{\eta (q) q}{\overline{{\eta }}(q)} \end{aligned}$$

(36)

where $\bar{{\eta }}\equiv 1-\eta$ denotes the share of income spent on good $Y$, as a function of the domestic consumer price $q$. Since in equilibrium consumption must exhaust the total real income $I^{h}$ of Home, i.e. $C^{Xh}+ p^{*}C^{Yh}= I^{h}= Q^{Xh}+ p^{*}Q^{Yh}$, we obtain:

$$\begin{aligned} C^{Yh} =\frac{\overline{{\eta }} I^{h}}{\eta q+\overline{{\eta }}p^{*}} \end{aligned}$$

(37)

Using $M^{Yh} = C^{Yh} -Q^{Yh}$, one can simplify $M^{Yh}$ to

$$\begin{aligned} M^{Yh} =\frac{\overline{{\eta }} Q^{Xh}-\eta q Q^{Yh}}{\eta q+\overline{{\eta }}p^{*}} \end{aligned}$$

(38)

where the RHS can be expressed completely in terms of $p^{*}$ and $\sigma$, since $q$ and $p$ are dependent via Eqs. (28) and (29). Calculating the derivative $\frac{\partial }{\partial \sigma }$ and collecting terms leads to:

$$\begin{aligned} \frac{\partial M^{Yh}}{\partial \sigma }=\frac{-\left[ {\frac{\partial Q^{Yh} }{\partial p} \frac{\partial p}{\partial \sigma }\left( {\eta q+\overline{{\eta }}p^{*}} \right) \left( {\eta q+\overline{{\eta }}p} \right) +\frac{\partial q}{\partial \sigma } \left( {\eta \overline{{\eta }}+q\frac{\partial \eta }{\partial q}} \right) I^{h}} \right] }{\left( {\eta q+\overline{{\eta }}p^{*}} \right) ^{2}} \end{aligned}$$

(39)

The derivative of $\eta$ with respect to $q$ is connected to the elasticity of substitution $\Sigma$ of $U^{h}$ by $\frac{\partial \eta }{\partial q}=\frac{\left( {\Sigma -1} \right) \eta \bar{{\eta }}}{q}$, leading to the final expression:

$$\begin{aligned} \frac{\partial M^{Yh}}{\partial \sigma }=\frac{-\left[ {\frac{\partial Q^{Yh} }{\partial p} \frac{\partial p}{\partial \sigma }\left( {\eta q+\overline{{\eta }}p^{*}} \right) \left( {\eta q+\overline{{\eta }}p} \right) +\frac{\partial q}{\partial \sigma } \eta \overline{{\eta }} \Sigma I^{h}} \right] }{\left( {\eta q+\overline{{\eta }}p^{*}} \right) ^{2}} \end{aligned}$$

(40)

While the denominator is always positive, both terms of the numerator can be either positive or negative, depending on the sign of $\frac{\partial p}{\partial \sigma }$ and $\frac{\partial q}{\partial \sigma }$. Since $\Sigma > 0$, and $\frac{\partial Q^{Yh} }{\partial p}$ is positive as given by Eq. (14), we have the following three cases: (i) Eq. (40) and hence Eq. (35) for $\frac{dp^{*}}{d\sigma }$ are positive, i.e. $p^{*}$ increases when $\sigma$ increases, if $\frac{\partial p}{\partial \sigma }<$ 0 and $\frac{\partial q}{\partial \sigma }<$ 0 (or if one term is negative and the other term is zero); likewise, Eq. (40) is negative and hence $p^{*}$ decreases if $\frac{\partial p}{\partial \sigma }\!>\! 0$ and $\frac{\partial q}{\partial \sigma } > 0$ (or if one term is positive and the other term is zero); (ii) the impact of $\sigma$ on $p^{*}$ is ambiguous if $\frac{\partial p}{\partial \sigma }$ and $\frac{\partial q}{\partial \sigma }$ have different signs. To determine the signs of $\frac{\partial p}{\partial \sigma }$ and $\frac{\partial q}{\partial \sigma }$, we obtain from Eq. (29):

$$\begin{aligned} \frac{\partial q}{\partial \sigma }=\gamma ^{Yf} -p^{*} \gamma ^{Xh} \end{aligned}$$

(41)

while combining Eqs. (28) and (29) yields for $p$ and $\frac{\partial p}{\partial \sigma }$:

$$\begin{aligned} p=p^{*}+\sigma \frac{\left( { \gamma ^{Yf} -\gamma ^{Yh} } \right) }{\left( {1-\sigma \gamma ^{Xh} } \right) }\Rightarrow \quad \frac{\partial p}{\partial \sigma }=\frac{\left( { \gamma ^{Yf} -\gamma ^{Yh} } \right) }{\left( {1-\sigma \gamma ^{Xh} } \right) ^{2}} \end{aligned}$$

(42)

Therefore, case (i) will hold if both of Home’s sectors are either more or less emission-intensive than Foreign’s export sector $Y$, respectively, and case (ii) otherwise. In the symmetric case (iii), i.e. when $\gamma ^{Xh} =\gamma ^{Xf} \text{ and } \gamma ^{Yh} =\gamma ^{Yf}$, $\frac{\partial p}{\partial \sigma }$ will become zero and Eq. (40) will simplify so that the sign of $\frac{dp^{*}}{d\sigma }$ only depends on Eq. (41) for $\frac{\partial q}{\partial \sigma }$. Accordingly, $\frac{dp^{*}}{d\sigma }$ will be positive if the $Y$-sector is less emission-intensive than the $X$-sector and negative if it is more emission-intensive.$\square$

Proof of Lemma 4

The proof proceeds analogously to the one of Lemma 3. Totally differentiating the market clearing condition $M^{Yh} (p^{*},\phi )-E^{Yf} (p^{*})=0$ yields:

$$\begin{aligned} \frac{dp^{*}}{d\phi } = -\frac{\frac{\partial M^{Yh} }{\partial \phi }}{\frac{\partial M^{Yh} }{\partial p^{*}}-\frac{\partial E^{Yf} }{\partial p^{*}}} \end{aligned}$$

(43)

As in Eq. (38), Home’s imports can then again be expressed as:

$$\begin{aligned} M^{Yh} =\frac{\overline{{\eta }} Q^{Xh}-\eta q Q^{Yh}}{\eta q+\overline{{\eta }}p^{*}} \end{aligned}$$

(44)

with $q$, $p$, and $p^{*}$ depending on $\phi$. Using Eq. (21), to express $p$ in terms of $q$ (and $\mu$, which is constant), and Eq. (29) to express $q$ in terms in terms of $p^{*}$ and $\phi$ the RHS of Eq. (42) can be written as a function of $p^{*}$ and $\phi$.¹⁹ Therefore, in analogy to Eq. (40), the sign of Eq. (43) is determined by the signs of $\frac{\partial p}{\partial \phi }$ and $\frac{\partial q}{\partial \phi }$:

$$\begin{aligned} \frac{\partial M^{Yh}}{\partial \phi }=\frac{-\left[ {\frac{\partial Q^{Yh} }{\partial p} \frac{\partial p}{\partial \phi }\left( {\eta q+\overline{{\eta }}p^{*}} \right) \left( {\eta q+\overline{{\eta }}p} \right) +\frac{\partial q}{\partial \phi } \eta \overline{{\eta }} \Sigma I^{h}} \right] }{\left( {\eta q+\overline{{\eta }}p^{*}} \right) ^{2}} \end{aligned}$$

(45)

As in Eq. (41), $\frac{\partial q}{\partial \phi }$ follows directly from Eq. (29):

$$\begin{aligned} \frac{\partial q}{\partial \phi }=\gamma ^{Yf} -p^{*} \gamma ^{Xh} . \end{aligned}$$

(46)

In order to determine $\frac{\partial p}{\partial \phi }$, we use Eqs. (21) and (29) to obtain:

$$\begin{aligned} p-p^{*}=\phi (\gamma ^{Yf} -p^{*} \gamma ^{Xh} )-\mu (\gamma ^{Yh} -p \gamma ^{Xh} ) \Rightarrow \frac{\partial p}{\partial \phi }=\frac{\gamma ^{Yf} -p^{*} \gamma ^{Xh} }{1-\mu \gamma ^{Xh} } \end{aligned}$$

(47)

As the denominator of Eq. (47)—which represents the normalized price of good $X$ minus the price of its ‘embodied’ emissions—is positive, $\frac{\partial p}{\partial \phi }$ and $\frac{\partial q}{\partial \phi }$ bear identical signs. Hence, if Home’s imports are more (less) carbon-intensive than its exports—i.e. it is a net importer (exporter) of emissions—Eq. (43) is negative (positive).$\square$

Proof of Lemma 5

The proof is basically identical to the one for Lemma 4. By substituting $\gamma ^{Yh}$ for $\gamma ^{Yf}$ in Eqs. (46) and (47), we obtain:

$$\begin{aligned} \frac{\partial q}{\partial \phi }=\gamma ^{Yh} -p^{*} \gamma ^{Xh} \end{aligned}$$

(48)

and

$$\begin{aligned} \frac{\partial p}{\partial \phi }=\frac{\gamma ^{Yh} -p^{*} \gamma ^{Xh} }{1-\mu \gamma ^{Yh} } \end{aligned}$$

(49)

Therefore, under the best available technology approach, Eq. (43) will be negative (positive) if Home’s export sector is more carbon-intensive than its non-export sector.$\square$

Proof of Proposition 6

As demonstrated in Sect. 4, Home’s welfare is maximized by setting a domestic production tax $\tau ^{opt}$ and a tariff $\theta ^{opt,}$ and, according to Eq. (21), this optimal production tax is equivalent to a domestic carbon price $\mu =-q^{Z}$. Therefore, given that $\tau ^{opt}$ is in place, the optimal tariff $\theta ^{opt }$ has to fulfill the following conditions:

$$\begin{aligned} \left. {\frac{dW^h }{d\theta }} \right| _{\theta =\theta ^{opt}} =0 \text{ and } \left. {\frac{d^{2}W^h }{d\theta ^{2}}} \right| _{\theta =\theta ^{opt}} <0 \end{aligned}$$

(50)

Assuming the welfare maximum to be unique, it directly follows that for any two tariffs $\theta ^{A},\, \theta ^{B}$ that deviate from the optimal tariff in the same direction—i.e. for which $sgn (\theta ^{A},- \theta ^{opt}) = sgn (\theta ^{B},- \theta ^{opt})$—the following has to hold:

$$\begin{aligned} sgn\left\{ {\left. {W^{h}} \right| _{\theta =\theta ^{A}} -\left. {W^{h}} \right| _{\theta =\theta ^{B}} } \right\} = -sgn\left\{ {\left| {\theta ^{A}-\theta ^{opt}} \right| -\left| {\theta ^{B}-\theta ^{opt}} \right| } \right\} \end{aligned}$$

(51)

Obviously, for the case in which the tariff $\theta ^{cons}$ implied by consumption-based pricing of emissions (Eq. 29) has the opposite sign of the optimal tariff, and given that under production-based emission pricing the tariff $\theta ^{prod}$ is zero (Eq. 20), Eq. (51) directly yields:

$$\begin{aligned} sgn\left\{ {\left. {W^{h}} \right| _{\theta =\theta ^{cons}} -\left. {W^{h}} \right| _{\theta =0} } \right\} = -1, \text{ i.e. } \left. {W^{h}} \right| _{\theta =\theta ^{cons}} < \left. {W^{h}} \right| _{\theta =0} \end{aligned}$$

(52)

which proves the first part of the proposition.

The second part of the proposition results from the observation that without specific assumptions regarding the functional form of $W ^{h}$, no general conclusions can be drawn for any two tariffs $\theta ^{A}\,\text{ and }\,\theta ^{B}$ that deviate from the optimal tariff in different directions—i.e. for which $sgn (\theta ^{A}- \theta ^{opt}) = -sgn (\theta ^{B}- \theta ^{opt})$.$\square$