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Adapting to Climate Change: Equilibrium Welfare Implications for Large and Small Economies

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. 1.

    All data are from the World Bank’s World Development Indicators 2013. GDP is calculated in current US dollars.

  2. 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. 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. 4.

    Of course, welfare cannot be higher when adaptation is unavailable if there is only one country. In that special case, \(S=y_i^2\) and neither condition (15) nor condition (16) can ever hold.

  5. 5.

    Note that \(Q=YS\) if there is only one country, so conditions (18) and (20) can never hold in that case.

  6. 6.

    We are grateful to an anonymous referee for drawing our attention to this point and its implications.

References

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Author information

Correspondence to Peter Kennedy.

Appendix

Appendix

Proof of Proposition 1

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

$$\begin{aligned} 2x_i y_i =y_i \delta \sum _{j=1}^n {y_j (1-a_j )} \quad \forall i \end{aligned}$$
(22)

and

$$\begin{aligned} 2\theta a_i y_i =y_i \delta \sum _{j=1}^n {y_j (1-x_j )} \quad \forall i \end{aligned}$$
(23)

From (23) we obtain

$$\begin{aligned} a_i =\frac{\delta }{2\theta }\sum _{j=1}^n {y_j (1-x_j )} \equiv a \quad \forall i \end{aligned}$$
(24)

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

$$\begin{aligned} x_i =\frac{\delta (1-a)Y}{2}\equiv x \quad \forall i \end{aligned}$$
(25)

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

$$\begin{aligned} a^{**}=\frac{\delta }{2\theta }\left( {1-\frac{\delta (1-a^{**})Y}{2}} \right) Y \end{aligned}$$
(26)

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

$$\begin{aligned} a^{**}=\frac{\delta (2-\delta Y)Y}{4\theta -\delta ^{2}Y^{2}} \end{aligned}$$
(27)

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

$$\begin{aligned} x^{**}=\frac{\delta (2\theta -\delta Y)Y}{4\theta -\delta ^{2}Y^{2}} \end{aligned}$$
(28)

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,

$$\begin{aligned} 2x_i y_i =\delta y_i^2 (1-a_i ) \end{aligned}$$
(29)

and

$$\begin{aligned} 2\theta a_i y_i =\delta (y_i (1-x_i )+E_{-i} )y_i \end{aligned}$$
(30)

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

$$\begin{aligned} x_i (E_{-i} )=\frac{\delta (2\theta -\delta y_i -\delta E_{-i} )y_i }{4\theta -\delta ^{2}y_i^2 } \end{aligned}$$
(31)

and

$$\begin{aligned} a_i (E_{-i} )=\frac{\delta (2y_i +2E_{-i} -\delta y_i^2 )}{4\theta -\delta ^{2}y_i^2 } \end{aligned}$$
(32)

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

$$\begin{aligned} e_i =[1-x_i (E_{-i} )]y_i=\frac{(4\theta -2\theta \delta y_i +\delta ^{2}y_i E_{-i} )}{4\theta -\delta ^{2}y_i^2 } \end{aligned}$$
(33)

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

$$\begin{aligned} e_i=y_i-\frac{(2\theta -\delta E)\delta y^{2}_{i}}{4\theta } \end{aligned}$$
(34)

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

$$\begin{aligned} \hat{{E}}=\frac{2\theta (2Y-\delta S)}{4\theta -\delta ^{2}S} \end{aligned}$$
(35)

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

$$\begin{aligned} \hat{{e}}_i =\frac{[4\theta -\delta ^{2}S-y_i (2\delta \theta -\delta ^{2}Y)]y_i }{4\theta -\delta ^{2}S} \end{aligned}$$
(36)

Then from (36) we can obtain

$$\begin{aligned} \hat{{x}}_i =1-\frac{\hat{{e}}_i }{y_i }=\frac{\delta (2\theta -\delta Y)y_i }{4\theta -\delta ^{2}S} \end{aligned}$$
(37)

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

$$\begin{aligned} \hat{{a}}_i =\frac{\delta (2Y-\delta S)}{4\theta -\delta ^{2}S} \quad \forall i \end{aligned}$$
(38)

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

$$\begin{aligned} \phi =\frac{2\delta \theta -\delta ^{2}Y}{4\theta -\delta ^{2}S}>0 \end{aligned}$$
(39)

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

Proof of Proposition 3

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

$$\begin{aligned} C(y_i ,\theta )=x_i^2 y_i +\theta a_i^2 y_i +\delta Ey_i (1-a_i ) \end{aligned}$$
(40)

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 \(\phi \):

$$\begin{aligned} C(y_i ,\theta )=\phi ^{2}y_i^3 +S\left( {\phi +\frac{\delta }{2}} \right) \left( {\frac{Y}{S}-\phi } \right) y_i \end{aligned}$$
(41)

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

$$\begin{aligned} \frac{Y}{S}-\phi =\left( {\frac{2\theta }{\delta S}} \right) \hat{{a}} \end{aligned}$$
(42)

so it must be positive if the equilibrium is interior.

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

$$\begin{aligned} \frac{\partial C(y_i ,\theta )}{\partial \theta }=2\phi y_i (y_i^2 -\tilde{s})\frac{\partial \phi }{\partial \theta } \end{aligned}$$
(43)

where

$$\begin{aligned} \tilde{s}=S-\frac{2Y-\delta S}{4\phi } \end{aligned}$$
(44)

and

$$\begin{aligned} \frac{\partial \phi }{\partial \theta }=\frac{2\delta ^{2}(2Y-\delta S)}{(4\theta -\delta ^{2}S)^{2}}>0 \end{aligned}$$
(45)

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

$$\begin{aligned} {\begin{array}{l} {\lim } \\ {\theta \rightarrow \infty } \\ \end{array} }\left( \phi \right) =\frac{\delta }{2} \end{aligned}$$
(46)

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

$$\begin{aligned} C_0 (y_i )=\left( {\frac{\delta ^{2}}{4}} \right) y_i^3 +\delta \left( {\frac{2Y-\delta S}{2}} \right) y_i \end{aligned}$$
(47)

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

$$\begin{aligned} \bar{{s}}\equiv S-\frac{2Y-\delta S}{2\phi +\delta } \end{aligned}$$
(48)

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

$$\begin{aligned} \tilde{s}-\bar{{s}}=\left( {\phi -\frac{\delta }{2}} \right) \left( {\frac{2Y-\delta S}{4\phi ^{2}+2\phi \delta }} \right) \end{aligned}$$
(49)

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:

$$\begin{aligned} G(Y,\theta )=\phi ^{2}Q+S\left( {\phi +\frac{\delta }{2}} \right) \left( {\frac{Y}{S}-\phi } \right) Y \end{aligned}$$
(50)

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

$$\begin{aligned} \frac{\partial G(Y,\theta )}{\partial \theta }=2\phi (Q-\tilde{Q})\frac{\partial \phi }{\partial \theta } \end{aligned}$$
(51)

where

$$\begin{aligned} \tilde{Q}=YS-\frac{(2Y-\delta S)Y}{4\phi } \end{aligned}$$
(52)

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:

$$\begin{aligned} G_0 (Y)=\left( {\frac{\delta ^{2}}{4}} \right) Q+\delta \left( {\frac{2Y-\delta S}{2}} \right) Y \end{aligned}$$
(53)

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

$$\begin{aligned} \bar{{Q}}=YS-\frac{(2Y-\delta S)Y}{2\phi +\delta } \end{aligned}$$
(54)

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

$$\begin{aligned} \tilde{Q}-\bar{{Q}}=Y\left( {\phi -\frac{\delta }{2}} \right) \left( {\frac{2Y-\delta S}{4\phi ^{2}+2\phi \delta }} \right) \end{aligned}$$
(55)

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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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