## Abstract

We show that the availability of adaptation can be welfare-reducing in the non-cooperative equilibrium in a setting with multiple countries. Adaptation is a private good while abatement is a public good. This means that substitution out of abatement and into adaptation by any one country imposes a negative externality on all other countries. The potentially deleterious impact of adaptation is asymmetric: small economies are most likely to be hurt by the availability of adaptation because they control a small fraction of global emissions relative to the biggest emitters.

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## Notes

- 1.
All data are from the World Bank’s

*World Development Indicators 2013*. GDP is calculated in current US dollars. - 2.
Some defensive measures (such as geoengineering) may have significant spillover effects on other countries, possibly negative; see Barrett (2008b). Here we restrict attention to purely private defensive measures.

- 3.
Ebert and Welsch (2011 and 2012) derive a more a general result. They assume that damage is strictly convex in global emissions, and this means that emissions are strategic substitutes when adaptation is not available; the best-response functions are negatively-sloped. The introduction of adaptation then creates the

*possibility*that emissions become strategic complements but the slopes of the best-response functions depend on the convexity of the damage function relative to the effectiveness of adaptation. We have assumed a linear damage function here because it allows us to derive closed-form solutions for the equilibrium while still highlighting the impact of adaptation on the nature of the strategic interaction between countries. - 4.
- 5.
- 6.
We are grateful to an anonymous referee for drawing our attention to this point and its implications.

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## Author information

## Appendix

### Appendix

### Proof of Proposition 1

The first-order conditions for \(x_i\) and \(a_i\) are, respectively,

and

From (23) we obtain

Thus, adaptation is identical across countries. Substituting \(a\) from (24) for \(a_j\) in (22), and rearranging, we obtain

Thus, technologies are identical across countries. Substituting \(x\) from (25) for \(x_j \) in (24), and rearranging, we obtain

Solving for \(a^{**}\) yields

Finally, substituting \(a^{**}\) for \(a\) in (25) yields

Solving for the conditions under which \(a^{**}\in [0,1]\) and \(x^{**}\in [0,1]\) yields the three parts of Proposition 1. It is straightforward to show that second-order conditions are satisfied. \(\square \)

### Proof of Proposition 2

The first-order conditions for \(x_i\) and \(a_i\) are, respectively,

and

Solving (29) and (30) yields best-response functions for \(x_i\) and \(a_i\). These are, respectively,

and

From (31) we can obtain the best-response function in terms of emissions:

Setting \(E_{-i} = E-e_i\) in (33) and rearranging to make \(e_i\) the subject, we have

Summing across *i* allows us to solve for the equilibrium \(E\):

Setting \(E=E^{\wedge } \) in (34), we can then solve for the equilibrium \(\hat{{e}}_i \):

Then from (36) we can obtain

Substituting \(\hat{{x}}_i\) for \(x_i\) in (29) then allows us to solve for \(a_i\):

Assumption 1 (from Sect. 4) ensures that these solutions are interior, that second-order conditions hold, and that the equilibrium is stable. We can then express \(\hat{{e}}_i =(1-\hat{{x}}_i )y_i \) as \(\hat{{e}}_i =y_i -\phi y_i^2 \), where

as reported in the text.\(\square \)

### Proof of Proposition 3

(a) Equilibrium domestic cost for country \(i\) is

In an interior equilibrium, \(x_i , a_i\) and \(E\) are evaluated at their equilibrium values given by (37), (38) and (35) respectively. It will prove useful to express this cost function in terms of \(\phi \):

where \(\phi \) is given by (39). The second bracketed term can be expressed as

so it must be positive if the equilibrium is interior.

Differentiating \(C(y_i ,\theta )\) with respect to \(\theta \) yields

where

and

at an interior equilibrium. Thus, \(\partial C/\partial \theta >0\) for \(y_i^2 >\tilde{s}\) and \(\partial C/\partial \theta <0\) for \(y_i^2 <\tilde{s}\).

(b) Making adaptation universally unavailable is equivalent to taking the limit

Making this substitution for \(\phi \) in (41) yields cost for country \(i\) when adaptation is universally unavailable :

where the second bracketed term must be positive in an interior equilibrium. Setting \(C(y_i ,\theta )=C_0 (y_i )\) and solving for \(y_i^2 \) yields a critical threshold denoted

Taking the difference \(\tilde{s}-\bar{{s}}\) yields

This is strictly positive at an interior equilibrium for any finite \(\theta \) since \(\partial \phi /\partial \theta >0\). Thus, \(\bar{{s}}<\tilde{s}\). Since \(\partial C/\partial \theta <0\) for \(y_i^2 <\tilde{s}\) (by part (a) above), it follows that \(\partial C/\partial \theta <0\) at \(y_i^2 =\bar{{s}}\). That is, \(C(y_i ,\theta )\) crosses \(C_0 (y_i )\) at \(y_i^2 =\bar{{s}}\) from above. Thus, \(C_0 (y_i )<C(y_i ,\theta )\) for \(y_i^2 <\bar{{s}}\) and \(C_0 (y_i )>C(y_i ,\theta )\) for \(y_i^2 >\bar{{s}}\).\(\square \)

### Proof of Proposition 4

(a) Total cost for country \(i\) is given by (41). Summing across \(i\) yields total global cost:

where \(Q=\sum _{i=1}^N {y_i^3 } \). Differentiating \(G(Y,\theta )\) with respect to \(\theta \) yields

where

and \(\partial \phi /\partial \theta >0\) is given by (45). Thus, \(\partial G/\partial \theta >0\) for \(Q>\tilde{Q}\) and \(\partial G/\partial \theta <0\) for \(Q<\tilde{Q}\).

(b) Taking the limit of \(G(Y,\theta )\) as \(\theta \rightarrow \infty \) yields total global cost when adaptation is universally unavailable:

Setting \(G(Y,\theta )=G_0 (Y)\) and solving for \(Q\) yields a critical threshold denoted

Taking the difference \(\tilde{Q}-\bar{{Q}}\) yields

This is strictly positive at an interior equilibrium for any finite \(\theta \) since \(\partial \phi /\partial \theta >0\). Thus, \(\bar{{Q}}<\tilde{Q}\). Since \(\partial G/\partial \theta <0\) for \(Q<\tilde{Q}\) (by part (a) above), it follows that \(\partial G/\partial \theta <0\) at \(Q=\bar{{Q}}\). That is, \(G(Y,\theta )\) crosses \(G_0 (Y)\) at \(Q=\bar{{Q}}\) from above. Thus, \(G_0 (Y)<G(Y,\theta )\) for \(Q<\bar{{Q}}\) and \(G_0 (Y)>G(Y,\theta )\) for \(Q>\bar{{Q}}\).\(\square \)

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### Cite this article

Farnham, M., Kennedy, P. Adapting to Climate Change: Equilibrium Welfare Implications for Large and Small Economies.
*Environ Resource Econ* **61, **345–363 (2015). https://doi.org/10.1007/s10640-014-9795-7

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### Keywords

- Climate change
- Adaptation
- Heterogeneous countries