Proof to Proposition 3

Proof to 3 (i). The first term in (7) is labeled the technique effect: ^∂η_∂t^jh

jM_ij^h. It gives the fall in emissions per unit of output of sector-h varieties. This effect can be decomposed further as (∇^tje_j · A^hj + e_j · ∇^tjA^h_j)M_ij^h, where ∇ denotes the gradient vector. First, the price increase of fossil fuel energy induces a substitution toward cleaner energy in all sectors of country j, with ^∂e_∂t^hj

j = ^{(βhνj−1)ehj(·)qf uel}

q_j^{f uel} < 0 for all sectors h. Second, there is a

shift toward the use of cleaner intermediates in the input-output matrix,∇tjA_j < 0. Both effects reduce the carbon intensity of production in country j and hence the technique effect reduces the carbon content of imports of i from j.

Proof to 3 (ii). We must pin down the effect on the numbers of varieties produced. In line with the literature on the home market effect (see e.g. Feenstra, 2004, p. 163 ff.) we calculate the change in the number of varieties due to a cost increase instead of directly solving for N_j^h. Let u^h_ij be the demand per variety in sector h, which is just _N^dhij_h

j. The market clearing condition ensures that the production of a sector in country j is equal to the c.i.f. demand from all countries, i.e. including the part that melts away.

y_j^h = (σh− 1)f^h =X

i

(1 + αh)uijτij. (B.4) Since output is fixed by technology parameters, the right-hand side also has to be fix.

Let’s assume that u^h_ij is fix, that is a carbon tax will lead to changes at the extensive margin N_k^h not the intensive margin u^h_ij. Totally differentiating the consumption of a variety from j to i yields:

Pˆ_i^h = σ_h

σ − 1ˆc^h_j, (B.5)

where ˆz = ^dz_z denotes the growth rate of a variable z. The change in the price index is proportional to the increase in costs in a country if the assumption of fixed consumption per variety holds.

Next we check whether there is a change in the number of varieties produced that is consistent with this change in the price index. Totally differentiate the price index to obtain

)^1−σhN_k^h denotes the share of country k products in sector h of country i and these shares add up to one. Plug in (B.5) into (B.6) to obtain

X

k

φ^h_ikNˆ_k^h = (σ_h− 1)X

k

φ^h_ikˆc^h_j − σhcˆ^h_j. (B.7) This gives a system of K equations that can be expressed in matrix notation as

Φ^h

constitutes a solution to the equation system. Put differently an increase in costs indeed induces changes at the extensive not the intensive margin. A cost increase (for example due to a higher carbon tax) will lead to a relocation of firms.

Proof to 3 (iii). The second term in (7) is the scale or pollution haven effect: η_j^h∂M_∂t^ijh

j .

It shows how the h-sector exports of country j react to a stricter climate policy. More specifically This can be simplified further. We can express (B.5) alternatively with partial derivatives with respect to t_j as

Thus, the change in the price index and the costs in (B.10) cancel each other out. This leaves

Only the change in the location of firms – the extensive margin – matters for the carbon content of trade. Next, we have to determine the sign of ^∂N_∂t^jh

j .

Let φ^h_ik⁻¹ denote the entries of the inverse of Φ^h. This gives the total weight of country-k varieties in consumption of country i after considering all input-output relations.

Then we can write (B.9) as

Nˆ_j^h = (σ_h− 1 − σ^h φ^h_jj⁻¹)ˆc^h_j, (B.12) Nˆ_i^h = (σ_h− 1)ˆc^hi − σ^h φ^h_ij⁻¹ˆc^h_j ∀ i 6= j. (B.13)

A carbon tax leads to a cost increase in country j, ˆc^h_j > 0. If φ^h_jj⁻¹ > ^σh_σ⁻¹

h , this implies a decrease in the number of varieties country j produces, i.e. ˆN_j^h < 0. Country j loses competitiveness in sector h. Put differently there is a pollution haven effect, because stricter environmental regulation in a country causes its exports to fall. The change in the numbers of varieties ˆN_i^h other countries i produce will be positive if the cost-weighted fraction of country-j varieties in consumption ^ˆc

hj

ˆ

c^h_i φ^h_ij⁻¹ < ^σh_σ⁻¹

h .

The intuition of this finding is straightforward. The unilateral climate policy puts country j to a comparative disadvantage in the energy-intensive differentiated goods sector compared to the labor-intensive numeraire sector. Thus other countries will produce more of the differentiated goods and export them while country j produces more of the

homogeneous numeraire good. This sectoral scale effect feeds through to less carbon imports in country i from country j in all sectors h.

Proof to 3 (iv). The dislocation of firms is governed by cost increases, which are given by The cost increase will be highest for dirty sectors, i.e. sectors with a high β_h. In other words, the loss in competitiveness is highest for carbon-intensive sectors. This means there is a negative correlation between sectoral emission intensity and the change in sectoral imports due to a carbon tax increase. That said, on the aggregate bilateral level, there will be a change in the composition of M_ij toward cleaner sectors beside the fall in trade volume when the exporter j imposes a stricter climate policy.

Proof to 3 (v). The carbon intensity of country i’s sectoral imports from j is given by

E_ij^h

M_ij^h = η^h_j. As shown in the proof to proposition 3 (i) the sectoral emission coefficient falls with a rising carbon tax in the very country.

The carbon intensity of country i’s bilateral imports is given by ¯η_j = _M^Eij

ij = change of sectoral imports due to the tax increase andP

h(η_j^h− ¯η^j) = P

hη_j^h− h¯η^j = 0.

The technique effect is negative for every sector which implies that the aggregate technique effect, the first term in the denominator of (B.15), is also negative. The composition effect states that the sectoral emission coefficient and the change in the sectoral imports due to a

carbon tax are negatively correlated. Therefore also the second term in the denominator of (B.15) is negative. Taken together, a tax increase reduces the carbon intensity of bilateral imports.