Unequal Vulnerability to Climate Change and the Transmission of Adverse Effects Through International Trade

Abstract

In this paper, we consider the unequal distribution of climate change damages in the world and we examine how the underlying costs can spread from a vulnerable to a non-vulnerable country through international trade. To focus on such indirect effects, we treat this topic in a North–South trade overlapping generations model in which the South is vulnerable to the damages entailed by global pollution while the North is not. We show that the impacts of climate change in the South can be sources of welfare loss for northern consumers in both the long and the short run. In the long run, an increase in the South’s vulnerability can reduce the welfare in the North economy even in the case in which it improves the terms of trade of the North. In the short run, the South’s vulnerability can also represent a source of intergenerational inequity in the North. Therefore, we emphasize the strong economic incentives for non-vulnerable—and a fortiori less vulnerable—economies to reduce the climate change damages on more vulnerable countries.

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Notes

  1. 1.

    However, note that the issues addressed by Ollivier (2016) differ from ours. While we focus on the transmission of indirect effects of climate change across countries, she examines whether trade increases pollution and/or welfare given the unequal vulnerability of countries to pollution.

  2. 2.

    The paper does not deal with environmental regulation and hence leaves aside the Pollution Haven Hypothesis issue.

  3. 3.

    We could consider the case where the brown sector produces a composite good used to save and to consume. However, it would make the analysis much more complex without changing the results (as long as the elasticity of substitution between brown and green consumption goods is one).

  4. 4.

    We could consider the case where \(0<\gamma ^{N}<\gamma ^{S}\) but it would make the analysis much more complex without qualitatively changing the results.

  5. 5.

    Following Yenokyan et al. (2014), this assumption means that once the investment has been put in place, the resulting stock is not mobile. Note that this is the standard specification in the dynamic trade model with two tradable goods, since Oniki and Uzawa (1965).

  6. 6.

    Assuming that countries differ in time preferences, \(\beta \), would affect the relative capital scarcity of countries but would not change our results qualitatively.

  7. 7.

    The initial conditions on \(K_{t=0}^i\) and \(G_{t=0}\) imply that \(p^W_{g,t=0}\) is given.

  8. 8.

    Capital depreciates fully in one period (equivalent to 40 years) such that the stock in \(t+1\) is given only by the amount of savings in t. By combining (iii) and (iv), the capital stock in \(t+1\) corresponds thus to the brown good production in both countries: \(Y_{b,t}^N+\Psi (G_t)Y_{b,t}^S=K_{t+1}^N +K_{t+1}^S\).

  9. 9.

    The mechanisms linking the world relative price \(p_g^W\) to pollution are further detailed in Sect. 4.1.

  10. 10.

    This condition implies \(L_b^S(T,0)>L_b^N(T,0)\).

  11. 11.

    In absence of pollution damage \(\gamma =0\), the PP locus is a horizontal line defined by \(p^{max}\equiv {\bar{p}}^W_g \) and the dynamics of both pollution and the relative price are always increasing as long as the initial conditions on capital stock and pollution define an economy with \(p^W_0<{\bar{p}}^W_g\) and \(G_0<{\bar{G}}\).

  12. 12.

    If the North were vulnerable to the direct effects of climate change in our setting, an additional negative effect on wages and interest rates would exist. Thus, it would make the decrease in the welfare of successive generations even more likely.

  13. 13.

    Details are provided in “Appendix 4”.

  14. 14.

    For the rest of the analysis, we define \(\Psi (\gamma, {\bar{G}}(\gamma ))\equiv \Psi (\gamma )\).

  15. 15.

    These expressions are obtained from the capital market equilibrium in each country and Eq. (8) along the steady state.

  16. 16.

    Note that if we had assumed that the South has a comparative advantage in the green production in the absence of damages, \({\hat{\gamma }}\) would be equal to zero for all \(\beta \), and the ToT effect would always be positive.

  17. 17.

    We consider that an efficient allocation of resources in the North is given by the solution of a social planner program that maximizes the welfare of northern agents when all northern generations are treated equally (i.e., the social discount factor is equal to zero). In the absence of population growth, the optimal stationary capital ratio is such that the marginal product of capital is equal to 1.

  18. 18.

    In a recent study, Ayong Le Kama and Pommeret (2017) emphasize the importance of devoting money to the CDM in order to control emissions abroad and hence reinforce the efficiency of adaptation measures implemented in the domestic economy.

  19. 19.

    A fall in \(\theta \) reduces the damage by its negative impact on the long run stock of pollution. A fall in \(\gamma \) reduces the damage in the long run provided that the elasticity of long run pollution stock to \(\gamma \) is not too high.

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Acknowledgements

We would like to thank the co-editor Robert Elliott, and the two anonymous referees for their helpful comments on an earlier draft. We are also grateful to the participants at the conferences LAGV 2017, EAERE 2017, FAERE 2017, SAET 2018 and at the workshops on location choices and environmental economics (2017, Pau), on growth, environment and population (2017, Nanterre) and on environmental regulation, trade and innovation (2018, Créteil). Supports from ANR GREEN-econ (ANR-16-CE03-0005) are gratefully acknowledged.

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Appendix

Appendix

Appendix 1: Proof of Lemma 1

Using (6) and (14), we have

$$\begin{aligned}&L_{b,t}^N+\left( \frac{A_b^N}{A_b^S}\right) ^\frac{\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^S}{A_g^N} \right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t)L_{b,t}^S \nonumber \\&\quad = \frac{(1-\alpha _b)\beta }{(1+\beta )}\left( 1+\left( \frac{A_b^N}{A_b^S}\right) ^\frac{\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^S}{A_g^N} \right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t)\right). \end{aligned}$$
(28)

From the capital market equilibrium in each country and Eq. (8), we have

$$\begin{aligned} k_{b,t+1}^N&= \left( \frac{p_{g,t+1}^WA_g^N}{A_b^N}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}, \\ k_{b,t+1}^S&= \left( \frac{p_{g,t+1}^WA_g^S}{A_b^S}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}, \end{aligned}$$

and

$$\begin{aligned} K_{t+1}^N&= \frac{\beta }{1+\beta }(1-\alpha _b)A_b^N \left( \frac{p_{g,t}^WA_g^N}{A_b^N}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b}, \\ K_{t+1}^S&= \frac{\beta }{1+\beta }(1-\alpha _b)A_b^S\Psi (G_t)\left( \frac{p_{g,t}^WA_g^S}{A_b^S}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b}. \end{aligned}$$

Combining these equations with (10), we obtain

$$\begin{aligned} L_{b,t+1}^N=\frac{(1-\alpha _b)}{(\alpha _b-\alpha _g)}\left[ -\alpha _g+\frac{\alpha _b\beta }{1+\beta }(1-\alpha _g)A_b^N \left( \frac{A_b^N}{A_g^N}\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}\left( \frac{(p_{g,t}^W)^{\alpha _b}}{p_{g,t+1}^W}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b-1}\right] \end{aligned}$$

and

$$\begin{aligned} L_{b,t+1}^S=\frac{(1-\alpha _b)}{(\alpha _b-\alpha _g)}\left[ -\alpha _g+\frac{\alpha _b\beta }{1+\beta }(1-\alpha _g)A_b^S \left( \frac{A_b^S}{A_g^S}\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}\left( \frac{(p_{g,t}^W)^{\alpha _b}}{p_{g,t+1}^W}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b-1}\Psi (G_t)\right]. \end{aligned}$$

The key variable that guarantees world general equilibrium with international trade is the world relative price \(p_{g,t}^W\). Equation (28) can be written as

$$\begin{aligned} &-\frac{(1+\beta )\alpha _g}{\beta \alpha _b(1-\alpha _g)}+A_b^N \left( \frac{A_b^N}{A_g^N}\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}\left( \frac{(p_{g,t}^W)^{\alpha _b}}{p_{g,t+1}^W}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b-1} \\&\quad +\,\left( \frac{A_b^N}{A_b^S}\right) ^\frac{\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^S}{A_g^N}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t) \\&\qquad \left( -\frac{(1+\beta )\alpha _g}{\beta \alpha _b(1-\alpha _g)}+A_b^S \left( \frac{A_b^S}{A_g^S}\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}\left( \frac{(p_{g,t}^W)^{\alpha _b}}{p_{g,t+1}^W}\right) ^\frac{1}{\alpha _b-\alpha _g}\Psi (G_t)\Lambda _{b}^{\alpha _b-1}\right) \\&\quad = \frac{(\alpha _b-\alpha _g)}{\alpha _b(1-\alpha _g)}\left( 1+\left( \frac{A_b^N}{A_b^S}\right) ^\frac{\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^S}{A_g^N}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t)\right) \end{aligned} $$

and after simplifications

$$\begin{aligned}&A_b^N \left( \frac{A_b^N}{A_g^N}\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g} \left( \frac{(p_{g,t}^W)^{\alpha _b}}{p_{g,t+1}^W}\right) ^\frac{1}{\alpha _b-\alpha _g}\Lambda _{b}^{\alpha _b-1} \left( 1+\left( \frac{A_b^S}{A_b^N}\right) ^\frac{1-2\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^N}{A_g^S}\right) ^\frac{1-2\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t)^2\right) \\&\quad = \frac{\beta \alpha _b+\alpha _g}{\beta \alpha _b(1-\alpha _g)}\left( 1+\left( \frac{A_b^N}{A_b^S}\right) ^\frac{\alpha _g}{\alpha _b-\alpha _g}\left( \frac{A_g^S}{A_g^N}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Psi (G_t)\right). \end{aligned}$$

Lemma follows. \(\square \)

Appendix 2: Existence and Unicity of Steady State, Proof of Proposition 1

Combining Eqs. (19) and (20) provided in Lemma 2, a steady state equilibrium \({\bar{G}}\) has to satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} &{\mathcal {F}}_1({\bar{G}})={\mathcal {F}}_2({\bar{G}}) \\ &{\bar{G}}<\Omega, \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} {\mathcal {F}}_1(G)&\equiv \left( G\right) ^\frac{\alpha _b-\alpha _g}{\alpha _b}\left( \frac{\theta }{\eta }\frac{(1-\alpha _b)}{(\alpha _b-\alpha _g)}(A_b^N)^\frac{1}{1-\alpha _b} \right) ^{\frac{-\alpha _b+\alpha _g}{\alpha _b}} \\&\quad \times \left( \frac{(\beta \alpha _b+\alpha _g)\Upsilon (G)}{(1+\beta )}\left[ 1+\Psi (G)\left( \frac{B}{T}\right) ^{\frac{1}{\alpha _b-\alpha _g}}\right] -\alpha _g\left[ 1+\left( \frac{B^{\alpha _g}}{T^{\alpha _b}}\right) ^\frac{1}{\alpha _b-\alpha _g}\right] \right) ^{\frac{-\alpha _b+\alpha _g}{\alpha _b}} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}_2(G)\equiv \left( \frac{\beta \alpha _b+\alpha _g}{x\beta \alpha _b(1-\alpha _g)}\Upsilon (G)\right) ^\frac{\alpha _b-\alpha _g}{\alpha _b-1}. \end{aligned}$$

with

$$\begin{aligned} \Upsilon (G)=\frac{1+\Psi (G)\left( \frac{B^{\alpha _g}}{T^{\alpha _b}}\right) ^\frac{1}{\alpha _b-\alpha _g}}{1+(\Psi (G))^2\left( \frac{1}{B}\right) ^\frac{1-2\alpha _g}{\alpha _b-\alpha _g}\left( T\right) ^\frac{1-2\alpha _b}{\alpha _b-\alpha _g}}. \end{aligned}$$

The corresponding steady state relative price is given by:

$$\begin{aligned} {\bar{p}}_g^W={\mathcal {F}}_1({\bar{G}}). \end{aligned}$$

The threshold \(\Omega \) is given by \(\Omega =\min \{G_1,G_2\}\) with \(G_1\) and \(G_2\) such that \(L_b^S(T,G_1)=0\) and \(L_b^N(T,G_2)=1\). Thus, the condition \(G<\Omega \) guarantees that the equilibrium steady state is diversified, i.e. \(0<L_b^N, L_b^S<1\).

We examine the properties of both functions on the interval \((0,\Omega )\). Under Assumptions 13, \({\mathcal {F}}_1(G)\) is increasing, with \({\mathcal {F}}_1(0)=0\) and \({\mathcal {F}}_1(\Omega )>0\). Concerning the function \({\mathcal {F}}_2(G)\), we have:

$$\begin{aligned} \frac{{\text {d}} {\mathcal {F}}_2(G)}{{\text {d}} G}=-\left( \frac{\beta \alpha _b+\alpha _g}{x \alpha _b\beta (1-\alpha _g)(1-\alpha _b)}\right) (\Upsilon (G))^\frac{1-\alpha _g}{\alpha _b-1}\frac{{\text {d}} \Upsilon (G)}{{\text {d}} G}. \end{aligned}$$

We have to examine the sign of the term \(\frac{{\text {d}}\Upsilon (G)}{{\text {d}}G}\), whose expression is the following:

$$\begin{aligned} \frac{{\text {d}} \Upsilon (G)}{{\text {d}}G}=\frac{-\Psi '(G)}{\left[ 1+(\Psi (G)^2N\right] ^2}\times \left[ \Psi (G)^2MN+2\Psi (G)N-M\right]. \end{aligned}$$

with \(M=\left( \frac{B^{\alpha _g}}{T^{\alpha _b}}\right) ^\frac{1}{\alpha _b-\alpha _g}\) and \(N=\left( \frac{1}{B}\right) ^\frac{1-2\alpha _g}{\alpha _b-\alpha _g}\left( T\right) ^\frac{1-2\alpha _b}{\alpha _b-\alpha _g}\). The first part of the equation being positive, we focus on the second one. Under Assumption 3 and for \(G<\Omega \), the second term is also positive, meaning that \({\mathcal {F}}_2(G)\) is downward sloping on the interval \((0,\Omega )\), with \({\mathcal {F}}_2(0)>0\) and \({\mathcal {F}}_2(\Omega )>0\). The condition to have a long-term diversified equilibrium is thus that:

$$\begin{aligned} {\mathcal {F}}_1(\Omega )>{\mathcal {F}}_2(\Omega ). \end{aligned}$$
(29)

The value for \(\Omega \) is driven by \(\gamma \). From (4), (15), (17) and (18), an increase in \(\gamma \) causes \(\Omega \) to fall and hence causes \({\mathcal {F}}_1(\Omega )\) to fall and \({\mathcal {F}}_2(\Omega )\) to rise. When \(\gamma =0\), \(\Omega \) tends to infinity with \(\lim _{\Omega \rightarrow \infty }{\mathcal {F}}_1(\Omega )>\lim _{\Omega \rightarrow \infty }{\mathcal {F}}_2(\Omega )\). When \(\gamma \) tends to infinity, \(\Omega \) is equal to zero with \({\mathcal {F}}_1(0)<{\mathcal {F}}_2(0)\). As a result, there exists a threshold \({\bar{\gamma }}\) such that when \(\gamma <{\bar{\gamma }}\) the condition given by (29) holds. In this case, there exists a unique steady state: \({\bar{G}}\equiv {\bar{G}}(\gamma )\in (0,\Omega )\).

Appendix 3: Welfare Analysis: Short-Term Implications in the Non-vulnerable Economy

The indirect utility in the North is given by the following expression:

$$\begin{aligned} V^N(w^{N}_t,R^{N}_{t+1}, p^{W}_{g,t}, p^{W}_{g,t+1})=\ln \left( \frac{1}{1+\beta } \frac{w^{N}_t}{p^{W}_{g,t}}\right) +\beta \ln \left( \frac{\beta }{1+\beta } \frac{R^{N}_{t+1}}{p^{W}_{t+1}}w^{N}_t\right). \end{aligned}$$

We can express factor prices as functions of the relative price by using (9). In this way, we obtain a function that depends on the relative price only:

$$\begin{aligned} V^N(p^{W}_{g,t}, p^{W}_{g,t+1})= {\mathcal {C}}_1+\left( \frac{\alpha _g+ \alpha _b\beta }{\alpha _b-\alpha _g}\right) \ln \left( p^{W}_{g,t}\right) -\beta \left( \frac{1-\alpha _g}{\alpha _b-\alpha _g}\right) \ln \left( p^{W}_{g,t+1}\right), \end{aligned}$$

with

$$\begin{aligned} {\mathcal {C}}_1&= \ln \left( \frac{1}{1+\beta }\right) +\beta \ln \left( \frac{\beta }{1+\beta }\right) +(1+\beta )\ln \left[ (1-\alpha _b)A_b\left( \frac{A_g}{A_b}\right) ^\frac{\alpha _b}{\alpha _b-\alpha _g}\Lambda _b^{\alpha _b} \right] \\& \quad+\,\beta \ln \left[ (1-\alpha _b)A_b\left( \frac{A_g}{A_b}\right) ^\frac{\alpha _b-1}{\alpha _b-\alpha _g}\Lambda _b^{\alpha _b-1} \right]. \end{aligned}$$

As \(p^{W}_{g,t+1}\) is a function of \(p^{W}_{g,t}\) and \(G_t\), using Eq. (15), we can also express the indirect utility as

$$\begin{aligned} V^N(p^{W}_{g,t}, G_t)={\mathcal {C}}_2+\left( \alpha _g\frac{1+\alpha _b\beta }{\alpha _b-\alpha _g}\right) \ln \left( p^{W}_{g,t}\right) +\beta (1-\alpha _g)\ln (\Upsilon (G_t)), \end{aligned}$$

with

$$\begin{aligned} {\mathcal {C}}_2={\mathcal {C}}_1-\beta (1-\alpha _b)\ln \left( \frac{\beta \alpha _b(1-\alpha _g)}{\beta \alpha _b+\alpha _g}\right). \end{aligned}$$

To describe the dynamics of this model, we define the two loci \(\hbox {GG} \equiv \{(p_{g,t}^W, G_t){:}\,G_{t+1}=G_{t}=G \}\) and \(\hbox {PP} \equiv \{(p_{g,t}^W, G_t){:}\,p_{g,t+1}^W=p_{g,t}^W=p_g^W \}\). Using Lemma 1, we have:

$$\begin{aligned}&{\text{GG}\,\text{locus}}{:}\; p^W_g=\left( {G}\right) ^\frac{\alpha _b-\alpha _g}{\alpha _b}\left( \frac{\theta }{\eta }\frac{(1-\alpha _b)}{(\alpha _b-\alpha _g)}(A_b^N)^\frac{1}{1-\alpha _b}\right) ^{\frac{-\alpha _b+\alpha _g}{\alpha _b}} \nonumber \\&\quad \times \left( \frac{(\beta \alpha _b+\alpha _g)\Upsilon ({G})}{(1+\beta )}\left[ 1+\Psi ({G})\left( \frac{B}{T}\right) ^{\frac{1}{\alpha _b-\alpha _g}}\right] -\alpha _g\left[ 1+\left( \frac{B^{\alpha _g}}{T^{\alpha _b}}\right) ^\frac{1}{\alpha _b-\alpha _g}\right] \right) ^{\frac{-\alpha _b+\alpha _g}{\alpha _b}} \nonumber \\&\quad ={\mathcal {F}}_1(G), \nonumber \\&{\text {PP}\,\text{locus}}{:}\; p_g^W =\left( \frac{\beta \alpha _b+\alpha _g}{x\beta \alpha _b(1-\alpha _g)}\Upsilon ({G})\right) ^\frac{\alpha _b-\alpha _g}{\alpha _b-1}={\mathcal {F}}_2(G). \end{aligned}$$
(30)

The functions \({\mathcal {F}}_1(G)\) and \({\mathcal {F}}_2(G)\) are already defined in “Appendix 2”, to derive the existence of the steady state. The dynamics in the North economy can be depicted by a phase diagram whose phase lines are given by GG and PP loci. Under Assumptions 13, the term \(\Upsilon ({G})\) is increasing in G for \(G\in (0,\Omega )\) meaning that PP locus is downward sloping. Using the results provided in “Appendix 2”, GG locus is upward sloping for \(G\in (0,\Omega )\). Thus, the dynamics can be depicted in Fig. 2.

The transitional path depends on the initial conditions on the pollution stock \(G_{t=0}\equiv G_0\) and on the price \(p^W_{g,t=0}\equiv p^W_{g,0}\). The latter is given by the initial conditions on the capital stocks of North and South \(K_{0}^N\) and \(K_{0}^S\), and on pollution \(G_0\). Indeed, from (8) and (10), we have \(L_{b,t}^i\equiv L_{b}^i(K_t^i, p^W_{g,t})\). Replacing it in the equilibrium condition on the brown good market (14) and using (6), we have:

$$\begin{aligned} A_b^NL_{b}^N(K_t^N, p^W_{g,t})+ A_b^SL_{b}^S(K_t^S, p^W_{g,t})\Psi (G_t)=\frac{(1-\alpha _b)\beta }{1+\beta }\left( A_b^N+\Psi (G_t)A_b^S\right). \end{aligned}$$

From this equality, we have:

$$\begin{aligned} p_{g,0}^W=\Theta (\underset{+}{K_0^N}, \underset{+}{K_0^S}, \underset{+}{G_0}), \end{aligned}$$

with \(\Theta (0, K_0^S, G_0)>0\), \(\Theta (K_0^N, 0, G_0)>0\), \(\Theta (0, 0, G_0)=0\) and \(\Theta (K_0^N, K_0^S, 0)>0\). We focus on the case in which \(G_0<{\bar{G}}\), and \(p_{g,0}^W<p^{max}\) with \(p^{max}\) the relative price of good in the absence of pollution. In that case, we can observe a transitional path with a decreasing trend in \(p_g^W\) if the initial conditions on \(K^S\), \(K^N\) and G are such that the initial relative price \(p_{g,0}^W\) is above the PP locus that prevails along the steady state. Finally, we can summarize this situation by the following conditions:

$$\begin{aligned} \left( \frac{\beta \alpha _b+\alpha _g}{\beta \alpha _b(1-\alpha _g)} \Upsilon ({{\bar{G}}})\right) ^\frac{\alpha _b}{\alpha _b-1}&\equiv {\bar{p}}^W_g<\Theta (K_0^N, K_0^S, G_0)<\left( \frac{\beta \alpha _b+\alpha _g}{\beta \alpha _b(1-\alpha _g)}\Upsilon ({G_0}) \right) ^\frac{\alpha _b}{\alpha _b-1} \nonumber \equiv p^{max} \end{aligned}$$
(31)

and

$$\begin{aligned} G_0<{\bar{G}}. \end{aligned}$$

In the absence of damage (\(\gamma =0\)), we have \({\bar{p}}^W_g=p^{max}\), meaning that the condition (31) never holds. Under Assumptions 13 and \(\gamma <{\bar{\gamma }}\), \(\Upsilon (G)\) is increasing in G. For all initial values \(G_{0}<{\bar{G}}\), we thus have \({\bar{p}}^W_g<p^{max}\). Given the properties of the function \(\Theta (K_0^N, K_0^S,G_0)\) presented above, when \(G_{0}<{\bar{G}}\) there always exists a set of value (\(K^N_0, K^S_0\)) that satisfies inequality (31), such that we observe a decreasing trend in \(p_g^W\) over time.

Appendix 4: Welfare Analysis: Long-Term Implications in the Non-vulnerable Economy

The stationary indirect utility in the North is given by the following expression:

$$\begin{aligned} V(w^{N},R^{N}, p^{W}_{g})=\ln \left( \frac{1}{1+\beta } \frac{w^{N}}{p^{W}_{g}}\right) +\beta \ln \left( \frac{\beta }{1+\beta } \frac{R^{N}}{p^{W}}w^{N}\right) \equiv V^N. \end{aligned}$$
(32)

Let \(\omega ^{N}=w^{N}/p_g^W\), from (32) it follows that:

$$\begin{aligned} {Sign} \left[ {\text {d}}V^N\right] ={Sign}\left[ \left( \frac{{\text {d}}\omega ^{N}}{{\text {d}}p_g^W}+\frac{\beta }{1+\beta }\frac{\omega ^{N}}{R^{N}}\frac{{\text {d}}R^{N}}{{\text {d}}p_g^W}\right) {{\text {d}}}p_g^W\right]. \end{aligned}$$

Using the fact that

$$\begin{aligned} {\text {d}}\omega ^{N}/{\text {d}}p^{W}_{g}=(p^{W}_{g})^{-1}({\text {d}}w^{N}/{\text {d}}p^{W}_{g}-w^{N}/p^{W}_{g}) \end{aligned}$$

and

$$\begin{aligned} \frac{\beta }{1+\beta }w^{N}=K^{N}, \end{aligned}$$

we have:

$$\begin{aligned} {Sign}\left[ {\text {d}}V^N\right] ={Sign} \left[ \left( \frac{K^{N}}{R^{N}} \frac{{\text {d}}R^{N}}{{\text {d}}p_g^W}- \frac{w^{N}}{p_g^W}+\frac{{\text {d}}w^{N}}{{\text {d}}p_g^W}\right) {\text {d}}p_g^W \right]. \end{aligned}$$
(33)

In each country, the gross domestic product is equal to the total revenue of capital and workers:

$$\begin{aligned} \left[ Y_{b}^{N}+Y_{g}^{N}p_{g}^{W}\right] =w^{N}+R^{N}K^{N}. \end{aligned}$$
(34)

By differentiating this equation, we obtain:

$$\begin{aligned} Y_{g}^{N}{\text {d}}p_{g}^{W}&= {\text {d}}w^{N}+K^{N}{\text {d}}R^{N},\nonumber \\ \frac{{\text {d}}w^{N}}{{\text {d}}p_g^W}&= Y_{g}^{N}-\frac{K^{N} {\text {d}}R^{N}}{{\text {d}}p_g^W}. \end{aligned}$$
(35)

Combining (34) and (35), we obtain:

$$\begin{aligned} \frac{{\text {d}}w^{N}}{{\text {d}}p_g^W}=\frac{w^{N}+R^{N}K^{N}- Y_{b}^{N}}{p^{N}_g}-\frac{K^{N}{\text {d}}R^{N}}{{\text {d}}p_g^W}. \end{aligned}$$
(36)

Moreover, using Eq. (6), we have:

$$\begin{aligned} \frac{{\text {d}}w^{N}}{{\text {d}}p_g^W}=\frac{\alpha _{b}}{\alpha _{b}-\alpha _{g}}\frac{w^{N}}{p_g^W};\quad \frac{{\text {d}}R^{N}}{{\text {d}}p_g^W}=-\frac{1-\alpha _{b}}{\alpha _{b}-\alpha _{g}}\frac{R^{N}}{p_g^W}. \end{aligned}$$
(37)

Using (36) and (37), Eq. (33) can be written as:

$$\begin{aligned} {Sign }\left[ \frac{dV^N}{{\text {d}}p_g^W}\right] = {Sign}\left[ \frac{K^{N}}{R^{N}}\frac{{\text {d}}R^{N}}{{\text {d}}p_g^W}-\frac{w^{N}}{p_g^W}+\frac{w^{N}+R^{N}K^{N}- Y_{b}^{N}}{p^{N}_g}-\frac{K^{N}{\text {d}}R^{N}}{{\text {d}}p_g^W} \right] \end{aligned}$$

and after simplifications, we have:

$$\begin{aligned} {Sign }\left[ \frac{dV}{{\text {d}}p_g^W}\right] ={Sign }\left[ \frac{K^{N}}{R^{N}}\frac{{\text {d}}R^{N}}{{\text {d}}p_g^W}(1-R^{N})+\frac{R^{N}K^{N}- Y_{b}^{N}}{p^{N}_g} \right]. \end{aligned}$$

Using (37),

$$\begin{aligned} {Sign }\left[ \frac{dV^N}{{\text {d}}p_g^W}\right] ={Sign }\left[ \left( \frac{-(1-\alpha _b)(1-R^N)K^{N}}{\alpha _b-\alpha _g}+R^{N}K^{N}- Y_{b}^{N}\right) \times \frac{1}{p_g^W} \right]. \end{aligned}$$

Finally, this expression can be expressed as follows:

$$\begin{aligned} {Sign }\left[ \frac{dV^N}{{\text {d}}p_g^W}\right] ={Sign }\left[ \left( \frac{(1-\alpha _g)R^NK^{N}-K^N(1-\alpha _g)+K^N(\alpha _b-\alpha _g)}{\alpha _b-\alpha _g}- Y_{b}^{N} \right) \times \frac{1}{p_g^W} \right], \end{aligned}$$

and we obtain the indirect utility given by Eq. (25).

Appendix 5: Pattern of Trade Along the Steady State, Proof of Propositions 3 and 4

Using (26), the pattern of specialization along the steady state is the following:

  1. (a)

    When \(\Psi (\gamma )>\frac{(B)^\frac{1-\alpha _g}{\alpha _b-\alpha _g}}{\left( T\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}}, \) North is a net exporter of the green good.

  2. (b)

    When \(\Psi (\gamma )<\frac{(B)^\frac{1-\alpha _g}{\alpha _b-\alpha _g}}{\left( T\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}},\) North is a net exporter of the brown good.

The pattern of specialization depends on how the undamaged part of production \(\Psi (\gamma )\) responds to the South’s vulnerability \(\gamma \). For \(\gamma =0\), \(\Psi (\gamma )=1\) and the scenario (a) is observed. For \(\gamma ={\bar{\gamma }}\), under Assumption 3, North is perfectly specialized in the brown good and the scenario (b) is observed.

To examine more precisely the conditions that lead the North to specialize in the brown good, we have to pay particular attention to how the undamaged part of production \(\Psi (\gamma )\) evolves with \(\gamma \). From (4), we have:

$$\begin{aligned} \dfrac{\partial \Psi (\gamma )}{\partial \gamma }=-\left( G(\gamma )\right) ^2\frac{\Psi (\gamma )}{1+\gamma \left( G(\gamma )\right) ^2}\left( \varepsilon _{G/\gamma }+\frac{1}{2}\right), \end{aligned}$$

with \(\varepsilon _{G/\gamma }=\frac{\partial G(\gamma )}{\partial \gamma }\frac{\gamma }{ G(\gamma )}\). Thus, when \(|\varepsilon _{G/\gamma }|<1/2\), we have \(\dfrac{\partial \Psi (\gamma )}{\partial \gamma }<0\) and there exists a threshold \({\hat{\gamma }}(\beta )\in (0, {\bar{\gamma }})\) such that \(\Psi \left( {\hat{\gamma }}(\beta )\right) =\frac{(B)^\frac{1-\alpha _g}{\alpha _b-\alpha _g}}{\left( T\right) ^\frac{1-\alpha _b}{\alpha _b-\alpha _g}}\).

The way the damages from climate change affect the North along the steady state depends largely on capital accumulation, which is driven by agent’s time preference \(\beta \). We have thus defined the steady state value for pollution as \({\bar{G}}\equiv G(\gamma, \beta )\) and the threshold on \(\gamma \) as a function of \(\beta \). We have \(\frac{\partial G(\gamma, \beta )}{\partial \beta }>0\). This result is derived by using functions \({\mathcal {F}}_1\) and \({\mathcal {F}}_2\), provided in “Appendix 2”, whose intersection gives the steady state equilibrium \(G(\gamma, \beta )\). Following an increase in \(\beta \), \({\mathcal {F}}_1(G)\) shifts downward while \({\mathcal {F}}_2(G)\) shifts upward, meaning that the equilibrium value \(G(\gamma, \beta )\) increases. As \(G(\gamma, \beta )\) is an increasing function of \(\beta \), from (26), \({\hat{\gamma }}(\beta )\) decreases with \(\beta \). Then, the propositions follow. \(\square \)

Appendix 6: Proof of Proposition 5

The social welfare in the North, defined by a social planner that treats all northern generations equally, is maximized for an allocation that satisfies that the marginal product of capital is equal to one (i.e., \(R^N=1\)). Using (6) along the steady state, we obtain that a marginal product of capital equal to one implies \(k_{b}^N=\left( A_b^N \alpha _b\right) ^\frac{1}{1-\alpha _b}\). From the equilibrium on the brown good market along the steady state, \(K^N=Y_b^N\), and (1), we have:

$$\begin{aligned} K^N=\frac{A_b^N\left( k_b^{N}\right) ^{\alpha _b}(1-\alpha _b)}{A_b^N(1-\alpha _g)\alpha _b\left( k_b^{N}\right) ^{\alpha _b-1}+\alpha _g- \alpha _b}. \end{aligned}$$
(38)

The stock of capital that guarantees a steady state marginal product of capital equal to one in the North is thus:

$$\begin{aligned} K^N=A_b^N\left( A_b^N \alpha _b\right) ^\frac{\alpha _b}{1-\alpha _b}\equiv {\tilde{K}}^N. \end{aligned}$$

From (8), (13), (20) and the equilibrium on the capital market along the steady state, \(K^N=s^N\), the stationary competitive stock of capital is:

$$\begin{aligned} K^N=\frac{\beta }{1+\beta }(1-\alpha _b)\left( A_b^N\right) ^\frac{1}{1-\alpha _b} \left( \frac{\beta \alpha _b(1-\alpha _g)}{(\beta \alpha _b+\alpha _g)\Upsilon (G(\gamma, \beta ))}\right) ^\frac{\alpha _b}{1-\alpha _b}\equiv K^N(\gamma, \beta ). \end{aligned}$$

Under Assumptions 13 and for \(\gamma <{\bar{\gamma }}\) and \(|\varepsilon _{G/\gamma }|<1/2\), \(\Upsilon (G(\gamma, \beta ))\) increases with \(\gamma \) which implies \(\partial K^N/\partial \gamma <0\). Indeed, \(\Upsilon \) depends on \(\gamma \) only through the undamaged part of production \(\Psi \). From Assumptions 13 and for \(\gamma <{\bar{\gamma }}\), the term \(\Upsilon \) decreases with \(\Psi \) and for \(|\varepsilon _{G/\gamma }|<1/2\) the undamaged part of production falls with \(\gamma \) along the steady state.

The competitive stock of capital corresponds to those maximizing the social welfare in the North if and only if:

$$\begin{aligned} K^N(\gamma, \beta )={\tilde{K}}^N\; \Leftrightarrow \;\frac{\beta }{1+\beta }(1-\alpha _b)\left( \frac{\beta (1-\alpha _g)}{(\beta \alpha _b+\alpha _g)\Upsilon (G(\gamma, \beta ))}\right) ^\frac{\alpha _b}{1-\alpha _b}=1. \end{aligned}$$

Under Assumptions 13 and for \(\gamma <{\bar{\gamma }}\), there exists a unique value of \(\gamma \) that guarantees that the competitive stock of capital satisfied \(R^N=1\): \({\tilde{\gamma }}(\beta )\). When \(\gamma >{\tilde{\gamma }}(\beta )\), \(R^N>1\) and \(K^N<{\tilde{K}}^N\): there is under-accumulation of capital. Conversely when \(\gamma <{\tilde{\gamma }}(\beta )\), \(R^N<1\) and \(K^N>{\tilde{K}}^N\): there is over-accumulation of capital. Under Assumptions 13 and for \(\gamma <{\bar{\gamma }}\), we have the following properties for \({\tilde{\gamma }}(\beta )\): \({\tilde{\gamma }}(\beta )\) increases with \(\beta \) and \({\tilde{\gamma }}(0)<0\), meaning that there is under-accumulation of capital \(\forall \;\gamma \) when \(\beta =0\). Given the fact that \(K^N\) is decreasing with \(\gamma \), an increase in the South’s vulnerability makes the competitive solution further to the efficient allocation in the first case and closer in the second case.

Appendix 7: Discussion on the Welfare Implications of a Decrease in Damages on Northern Generations

We do not formalize policy measures in our study but we can appreciate their potential benefits by examining the impacts of a variation in parameters \(\theta \) and/or \(\gamma \). We can suppose an investment in clean development mechanisms (CDM) that aims at mitigating emissions (i.e., decreasing the pollution intensity \(\theta \)), or a contribution to climate funds supporting adaptation that consists in reducing the South’s vulnerability (i.e., \(\gamma \)).

Leaving aside the financial cost of such an environmental policy, we provide some insight into the necessary conditions for its acceptability by the different generations alive when the policy is first implemented in the North. To this aim, we examine the distributive effect of an exogenous fall in damages (captured by a decrease in \(\gamma \) and/or \(\theta \)) between consumption of young and old people at the steady state.Footnote 19

The following expressions correspond to the welfare—indirect utility—of the young (\(V^{Ny}(p^W_g)\)) and of the old (\(V^{No}(p^W_g)\)) northern agents that coexist along the steady state:

$$\begin{aligned} V^{Ny}(p^W_g)={\mathcal {C}}^y+\frac{\alpha _g}{\alpha _b-\alpha _g}\ln (p^W_g);\quad V^{No}(p^W_g)={\mathcal {C}}^o+\frac{\alpha _b+\alpha _g-1}{\alpha _b-\alpha _g}\ln (p^W_g), \end{aligned}$$
(39)

with \({\mathcal {C}}^y=\ln \left( A_b^N(1-\alpha _b)\Lambda _b^{\alpha _b}\left[ \frac{A_g^N}{A_b^N}\right] ^\frac{\alpha _b}{\alpha _b-\alpha _g}\right) \) and \({\mathcal {C}}^o=\ln \left( \left( {A_b^N}\right) ^2\alpha _b(1-\alpha _b)\Lambda _b^{2\alpha _b-1}\left[ \frac{A_g^N}{A_b^N}\right] ^\frac{2\alpha _b-1}{\alpha _b-\alpha _g}\right) \).

From the previous analysis, we know that an—exogenous—decrease in damages in the long run increases the relative price \( p_g^W\) (see Lemma 1 with Assumptions 13). Using the indirect utility functions given in (39), we obtain:

Result 1

Under Assumptions 13, let\(|\varepsilon _{G/\gamma }|<1/2\)and\(\gamma <{\bar{\gamma }}\). At the steady state, when the damages from climate change on the South decrease, the welfare of the northern young always increases while the welfare of the northern elderly decreases (resp. increases) for\(\alpha _b+\alpha _g<1\) (resp. \(\alpha _b+\alpha _g>1\)).

This result gives intuitions about how northern agents coexisting at a given period of time are affected by a reduction in environmental damages in the South. We reveal that it could create a potential conflict between the two generations—young and old—as they are not affected in the same way by the spillover effects of pollution.

As the damages in the South lower the return of labor, it necessarily decreases the first period utility of young northern agents. However, in the absence of direct damages from climate change in the North, the elderly benefits from the damages in the South when capital intensity is not too high (\(\alpha _b+\alpha _g<1\)). Under this condition, they have no economic incentive to accept and to contribute to the funding of an environmental policy aiming at proving assistance to the South. Even if such measures may generate benefits in the future, these benefits would occur beyond the lifespan of the elderly generation.

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Constant, K., Davin, M. Unequal Vulnerability to Climate Change and the Transmission of Adverse Effects Through International Trade. Environ Resource Econ 74, 727–759 (2019). https://doi.org/10.1007/s10640-019-00345-8

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Keywords

  • International trade
  • Climate change
  • Heterogeneous damages
  • Overlapping generations

JEL Classification

  • F18
  • F43
  • O41
  • Q56