Abstract
This paper develops a two-region (North and South) dynamic model in which regional stocks of effective labor are negatively influenced by the global stock of pollution. By characterizing the equilibrium strategy of each region we show that the regions’ best responses can be strategic complements through a dynamic complementarity effect. The model is then used to analyze the impact of adaptation assistance from North to South. It is shown that North’s unilateral assistance to South (thus enhancing South’s adaptation capacity) can facilitate pollution mitigation in both regions, especially when the assistance is targeted at labor protection. Pollution might increase in the short run, but in the long run the level of pollution will decline. The adaptation assistance we propose is incentive compatible and Pareto improving.
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We are grateful to Ingmar Schumacher, the editor, and four anonymous referees for their positive and constructive comments, and to Larry Karp, Christian Traeger, Rintaro Yamaguchi, and Akihiko Yanase for suggestions and discussions. Earlier versions of this paper were presented at seminars in Tilburg University and UC Berkeley, and at the EconomiX-CNRS workshop at Université Paris Ouest (December 2014), the EAERE Conference at Zurich (June 2016), and the Conference on Economics of Global Interactions at Bari (September 2016). Sakamoto acknowledges financial support from the Japan Society for the Promotion of Science (JSPS) and is grateful to the Department of Agricultural and Resource Economics at UC Berkeley for its hospitality.
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Appendix: Proofs of Propositions
Appendix: Proofs of Propositions
1.1 Proof of Proposition 1
1.1.1 Problem of Period \(t=2\)
Since \(V_{i,3}(Z_{3})=0\), it follows that \(E_{i,2}=\bar{E}_{i}\) for both regions. Hence, for each \(i \in \{n,s\}\),
and we find that the value function is
where \(Z_{2}=(L_{n,2},L_{s,2},M_{2},R_{2})\). We note that
and
1.1.2 Problem of Period \(t=1\)
Combining Eqs. (28) and (12), we may write the first-order condition as
which determines \((E_{n,1}, E_{s,1})\) as a function of \(Z_{1}=(L_{n,1},L_{s,1},M_{1},R_{1})\). Notice that the left-hand side is decreasing in \(E_{i,1}\) and is independent of \(E_{j,1}\) whereas the right-hand side is increasing in both \(E_{i,1}\) and \(E_{j,1}\). It follows that the equilibrium combination of \((E_{n,1},E_{s,1})\) satisfies the stability conditions of Dixit (1986).
Totally differentiating Eq. (29) yields
where we define
Notice that \(\lim _{\xi \rightarrow 1}\Psi _{i}=0\) and the substitution yields
The value function is
where \(E_{i,1}\) is implicitly defined by Eq. (29) and \(V_{i,2}\) is given by Eq. (26). Using Eqs. (27), (28), and (29), we obtain
where \(dE_{j,1}/dM_{1}\) is given by Eqs. (30), (31), and (32). Moreover,
where
and
Since \(\lim _{\xi \rightarrow 1}\Psi _{i}=0\), it follows that
and therefore
1.1.3 Problem of Period \(t=0\)
Combining Eqs. (37) and (12), we may write the first-order condition as
where \({\textit{MB}}_{i}(E_{i,0})\) is the marginal benefit from emission defined by
and \({\textit{MD}}_{i}(E_{i,0},E_{j,0})\) is the marginal damage from emission defined by
Notice that for any \(E_{j,0}\ge 0\), we have
and
Hence, for each \(E_{j,0}\ge 0\), there exists \(E_{i,0} > 0\) at which the graph of \({\textit{MB}}_{i}\) crosses the graph of \({\textit{MD}}_{i}\) from above. Let \(E_{i}^{*}(E_{j,0})>0\) denote the smallest of such points, which is the reaction function of region i. We characterize the equilibrium level of \(E_{s,0}\) by \(E_{s,0}=E^{*}_{s} (E_{n}^{*}(E_{s,0}))\), which is the smallest solution of
The equilibrium level of \(E_{n,0}\) is then determined by \(E_{n,0}=E_{n}^{*}(E_{s,0})\). We note that, for both regions, the second-order condition is satisfied because the marginal benefit curve crosses the marginal damage curve from above.
To see that this equilibrium is stable, let us define
for each \(i \in \{n,s\}\). The second-order condition implies
at equilibrium. Also, by the definition of \(E^{*}_{n}\), we have
and hence
Moreover, since the graph of \({\textit{MB}}_{i}\) crosses the graph of \({\textit{MD}}_{i}\) from above, we must have
at equilibrium. Combining Eqs. (50), (52), and (53) yields
meaning that the stability conditions of Dixit (1986) are satisfied.
1.2 Proof of Proposition 2
To characterize \(E_{i,0}\), observe first that the marginal benefit \({\textit{MB}}_{i}(E_{i,0})\) of emission is decreasing in \(E_{i,0}\). On the other hand, Eq. (42) implies that at least when \(\xi\) is in the neighborhood of \(\xi =1\), the marginal damage \({\textit{MD}}_{i}(E_{i,0},E_{j,0})\) of emission is decreasing in \(E_{j,0}\), meaning that the marginal damage curve of a region shifts upwards as the other region reduce its emission. Therefore, we conclude that there exists \(\bar{\xi }>1\) such that the short-run regional emissions are strategic complements as long as \(\xi < \bar{\xi }\).
1.3 Proof of Proposition 3
See text.
1.4 Proof of Proposition 4
Totally differentiating Eq. (43) with respect to R for both \(i \in \{n,s\}\) and evaluating every term at \(R=0\) yields
where
We first note that every element of matrix D is strictly positive: the diagonal elements because of the second-order condition and the off-diagonal elements because of Eqs. (45) and (42). The determinant of D is also strictly positive due to the stability condition (54).
Since every entry of matrix \(D/\det (D)\) is strictly positive, Eq. (55) indicates that \(dE_{i,0}/dR\) is a positive linear combination of \(\partial {\textit{MB}}_{n}/\partial R\) and \(-\partial {\textit{MD}}_{s}/\partial R\). We observe from Eq. (56) that \(\partial {\textit{MB}}_{n}/\partial R\) is strictly positive. This term represents the income effect we discussed in the main text. The other partial derivative \(\partial {\textit{MD}}_{s}/\partial R\) in Eq. (57) may be positive or negative, depending on the relative importance of substitution and complementarity effects.
Notice that in Eq. (57), the terms in square brackets are strictly positive if the damage parameter \(\delta _{s}^{L}\) is sufficiently large. A sufficient (but by no means necessary) condition for such a case to be true is
which is satisfied by a sufficiently large value of \(\delta _{s}^{L}\). Accordingly, under Assumption 1, \(\partial {\textit{MD}}_{s}/\partial R\) is a strictly increasing linear function of \(\varepsilon ^{L}\). On the other hand, the income effect, \(\partial {\textit{MB}}_{n}/\partial R\), is independent of \(\varepsilon ^{L}\), as is clear from Eq. (56). Hence, by Eq. (55), we know that \(dE_{i,0}/dR\) is a strictly decreasing linear function of \(\varepsilon ^{L}\). In particular, if we define
and
it follows that for each region i, \(dE_{i,0}/dR<0\) if and only if \(\varepsilon ^{L} > \varepsilon ^{L}_{E_{i,0}}\).
Finally, Eq. (54) implies
which, together with Eqs. (60) and (61), yields \(\varepsilon _{E_{n,0}}^{L} >\varepsilon _{E_{s,0}}^{L}\). For the sake of completeness, we also note that the two thresholds coincide if there is no income effect (i.e., if \(\partial {\textit{MB}}_{n}/\partial R = 0\)).
1.5 Proof of Proposition 5
First notice
where, by Proposition 4, the right-hand side is strictly decreasing in \(\varepsilon ^{L}\). Also, Proposition 4 shows that there exist thresholds \(\varepsilon ^{L}_{E_{n,0}}\) and \(\varepsilon ^{L}_{E_{s,0}}\) with \(0<\varepsilon _{E_{s,0}}^{L} <\varepsilon _{E_{n,0}}^{L}\) such that
and
Then there must exist \(\varepsilon ^{L}_{M_{1}}\) in the open interval \((\varepsilon ^{L}_{E_{s,0}}, \varepsilon ^{L}_{E_{n,0}})\) such that
Next, observe
where the second term on the right-hand side is zero. By Eq. (17), the third term is strictly negative and strictly increasing in \(\varepsilon ^{L}\). Therefore, combined with Eq. (66), this implies that there exists \(\varepsilon ^{L}_{M_{2}}<\varepsilon ^{L}_{M_{1}}\) such that
which completes the proof.
1.6 Proof of Proposition 6
By the envelope theorem, a marginal change in the state variables has no first-order effect on \(W_{i}(R)\) or \(V_{i,t}(Z_{t})\) through the region i’s own control, \(E_{i,t}\). Hence,
where
and
Therefore,
and
where we use \(\phi _{R}^{2}\approx 0\). Evaluating these equations at \(R=0\) yields Eqs. (21) and (22).
1.7 Proof of Proposition 7
Since \(A_{n}\) does not affect the equilibrium levels of emission (when evaluated at \(R=0\)), it follows from Eqs. (55) and (56) that for each \(i \in \{n,s\}\), \(dE_{i,0}/dR\) is strictly decreasing in \(A_{n}\). Then, for each given level of \(\varepsilon ^{L}\), Eqs. (21) and (22) show that \(dW_{i}/dR\) is strictly increasing in \(A_{n}\). Since \(dW_{i}/dR\) is strictly increasing in \(\varepsilon ^{L}\), this implies that the threshold \(\varepsilon _{W_{i}}^{L}\) is strictly decreasing in \(A_{n}\).
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Sakamoto, H., Ikefuji, M. & Magnus, J.R. Adaptation for Mitigation. Environ Resource Econ 75, 457–484 (2020). https://doi.org/10.1007/s10640-019-00396-x
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DOI: https://doi.org/10.1007/s10640-019-00396-x