Adaptation to Climate Change: Commitment and Timing Issues


We study the impact of timing and commitment on adaptation and mitigation policies in the context of international environmental problems. Adaptation policies present the characteristics of a private good and may require a prior investment, while mitigation policies produce a public good. In a stylized model, we evaluate the impact of strategic commitment and leadership considerations when countries with different attitudes towards environmental cooperation coexist. We obtain equilibrium abatement and adaptation levels and environmental costs under partial cooperation for various timing and leadership scenarios. Crucially, global environmental costs suffered by countries are found to be greater when adaptation measures can be used strategically.

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  1. 1.

    For a recent survey please refer to Agrawala et al. (2011).

  2. 2.

    Namely, the decision about the adaptation level is made before, after and simultaneously with the mitigation one, where the last two sequences are shown to be equivalent.

  3. 3.

    Other papers adopting stylized functional forms in the literature use slightly different assumptions. The model of Buob and Stephan (2011) is consistent with (2), but uses a Cobb–Douglas formulation for the players’ utility. On the other hand, both Farnham and Kennedy (2014) and Marrouch and Ray Chaudhuri (2011) assume that the damage cost is bi-linear, which requires additionnal conditions on parameter values to ensure that the optimization problems are convex and that marginal costs have the expected signs.

  4. 4.

    The alternative assumption that cooperating countries choose their mitigation levels jointly and their investment in adaptive measures individually is called semi-cooperation in Zehaie (2009).

  5. 5.

    Notice that, because of this strategic role, the equilibrium results would be different if cooperators agreed to coordinate only their mitigation policies (semi-cooperation).

  6. 6.

    However, it would be convenient for a group of collaborating countries to make the first move and become leaders.

  7. 7.

    Numerical investigations show that, under leadership, prior investment is inefficient at the aggregate level, even if it may happen that leaders suffer a lower cost than in the concurrent investment case.


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Correspondence to Michèle Breton.



In the following proofs, we use the auxiliary variables \(k\equiv \theta \omega -1>0\) and XY and W as defined in (12)–(14) to simplify the notation.

Claim 1


$$\begin{aligned} K_{1}\frac{a_{C}^{\textit{PN}}-a_{I}^{\textit{PN}}}{n\omega }=kW\left( Y-\omega \right) \left( X-\omega \right) >0. \end{aligned}$$

\(m_{C}^{\textit{PN}}\gtrless m_{I}^{\textit{PN}}\):

$$\begin{aligned} K_{1}\frac{m_{C}^{\textit{PN}}-m_{I}^{\textit{PN}}}{kn\omega W}=\theta \omega W\left( np-1\right) -XY+np\omega \left( X+Y-\omega \right) .\end{aligned}$$

This difference is negative if

$$\begin{aligned} 0<\theta <\frac{XY-np\omega \left( X+Y-\omega \right) }{\omega W\left( np-1\right) }=np\omega \frac{nq-1-np}{W}, \end{aligned}$$

which requires that \(nq-1>np\) (more defectors than cooperators).


$$\begin{aligned} 2K_{1}^{2}\frac{c_{C}^{\textit{PN}}-c_{I}^{\textit{PN}}}{n^{2}k^{2}W\omega \left( X-\omega \right) }= & {} kW\left( kW^{2}+\omega \left( 2X+\omega ^{2}+Y^{2}+X\left( 2X-\omega \right) +2Y\left( X-\omega \right) \right) \right) \\&+\,\left( \omega +1\right) \left( X+1\right) \left( \left( Y-\omega \right) \left( 3Y+X\omega +1\right) +Y^{2}\left( X-2\omega \right) \right) \\&+\,kW\left( 4\left( X+Y+XY\right) +X\left( 2X+Y^{2}\right) +2\left( Y^{2}+1\right) \right) \\&+\,\omega ^{2}\left( X+1\right) \left( \left( \omega +1\right) \left( Y+1\right) +X^{2}\right) \\&+\,\left( X+1\right) \left( 2\left( X+Y\right) +Y\left( 3X+Y^{2}\right) +X^{2}+1\right) \\&+\,\left( X+1\right) \left( \omega \left( 2\left( X+Y\right) +Y\left( 6X+Y^{2}\right) +2X^{2}+1\right) \right) \\> & {} 0. \end{aligned}$$

\(\square \)

Claim 2


$$\begin{aligned} K_{4}\left( m_{C}^{\textit{SN}}-m_{I}^{\textit{SN}}\right) =nk\left( np-1\right) >0. \end{aligned}$$

Total cost is higher for cooperating countries since the abatement and environmental costs are equal for both types of countries. \(\square \)

Claim 3


$$\begin{aligned} K_{5}\frac{a_{C}^{PL}-a_{I}^{PL}}{kn\omega }=-G^{2}U\omega \left( Y+1\right) k-U\omega W\left( G+U\omega \right) \left( Y+1\right) -GXY\left( G+U\omega \right) <0. \end{aligned}$$

Numerical investigations show that mitigation levels and total costs differences can be positive or negative. For mitigation levels, \(m_{C}^{PL}\le m_{I}^{PL}\) when

$$\begin{aligned} np\le \frac{\left( H+W\left( \omega +1\right) \left( \omega \left( Y+1\right) -G\right) \right) \left( \theta \omega H\left( Y+1\right) -XY\left( G+\omega U\right) \right) }{\left( U+\theta G\right) \left( -\omega UW+YG\left( Y+1\right) +\theta \omega \left( H-G^{2}\right) \right) G\omega }. \end{aligned}$$

\(\square \)

Claim 4

\(m_{C}^{\textit{SL}}\gtrless m_{I}^{\textit{SL}}\):

$$\begin{aligned} \omega K_{6}\frac{m_{C}^{\textit{SL}}-m_{I}^{\textit{SL}}}{kn}=k\left( np-nq\omega -1\right) +np-1 \end{aligned}$$

Since adaptation levels are the same for the two types of countries, both \(m_{C}^{\textit{SL}}<m_{I}^{\textit{SL}}\) and \(c_{C}^{\textit{SL}}<c_{I}^{\textit{SL}}\) hold when

$$\begin{aligned} np<1+\frac{k}{k+1}nq\omega . \end{aligned}$$

\(\square \)

Claim 5

  1. (a)


    $$\begin{aligned} 2K_{3}^{2}K_{5}^{2}\frac{c^{\textit{PI}}-c^{\textit{SI}}}{k^{2}n^{2}\left( n-1\right) ^{2}}= & {} k\left( \omega +2\right) \left( n\omega +1\right) ^{2}+\left( \omega +1\right) \left( \omega \left( 2n-1\right) +2\right)>0 \\ 2\omega K_{2}K_{5}^{2}\frac{c^{\textit{SI}}-c^{\textit{BF}}}{kn^{2}}= & {} k\left( n-1\right) ^{2}\left( k+1\right) >0. \end{aligned}$$
  1. (b)


    $$\begin{aligned} K_{3}K_{5}\left( a^{\textit{PI}}-a^{\textit{SI}}\right)= & {} nk\omega \left( n-1\right) \left( n\omega +1\right) >0 \\ K_{2}K_{5}\frac{a^{\textit{SI}}-a^{\textit{BF}}}{n}= & {} kn\left( n-1\right) . \end{aligned}$$
  2. (c)


    $$\begin{aligned} K_{3}K_{5}\frac{m^{\textit{PI}}-m^{\textit{SI}}}{kn}= & {} -\omega \left( n-1\right)<0 \\ K_{2}K_{5}\frac{m^{\textit{SI}}-m^{\textit{BF}}}{kn}= & {} -\left( n-1\right) \frac{k+1}{\omega }<0. \end{aligned}$$

\(\square \)

Claim 6

  1. (a)


    $$\begin{aligned} 2\omega K_{2}K_{4}^{2}\frac{c_{C}^{\textit{SN}}-c^{\textit{FB}}}{k^{2}n^{3}}=knq\omega \left( np-1\right) \left( q+np\left( p+1\right) \right) +q\left( n+np-2\right) \left( k+1\right) >0. \end{aligned}$$

    \(c^{\textit{FB}}>c_{C}^{\textit{PN}},\) \(c^{\textit{FB}}\gtrless c_{I}^{\textit{SN}}\) and \(c^{\textit{FB}}\gtrless c_{I}^{\textit{PN}}\) were checked numerically. For instance, for \(n=100,\) \(np=60,\) \(\omega =0.4\) and \(\theta =4\), the cost of the individualist countries is smaller than in the first best solution for both types of adaptation.

  1. (b)

    \(a^{\textit{FB}}<a_{C}^{\textit{SN}}=a_{I}^{\textit{SN}}\) is immediate since \(K_{4}=\theta +n\left( \left( 1-p\right) +np^{2}\right) k<\theta +kn^{2}=K_{2}.\) We already showed that \(a_{I}^{\textit{PN}}<a_{C}^{\textit{PN}}.\) \(a^{\textit{FB}}<a_{I}^{\textit{PN}}\) was checked numerically.

  2. (c)

    \(m^{\textit{FB}}>m_{I}^{\textit{SN}}\): this is immediate from \(K_{4}<K_{2}\).

    \(m^{\textit{FB}}>m_{I}^{\textit{PN}}\) was checked numerically.

    \(m^{\textit{FB}}\gtrless m_{C}^{\textit{PN}}\): for different set of parameters, we obtained both signs for the difference \(m^{\textit{FB}}-m_{C}^{\textit{PN}}\) and \(m^{\textit{FB}}-m_{C}^{\textit{SN}}\). The mitigation level of cooperating countries is higher than in the first best solution when the mitigation cost coefficient \(\gamma _{M}\) is large compared to the environmental sensitivity \(\gamma _{D}\).

\(\square \)

Claim 7

  1. (a)

    \(c_{C}^{\textit{PN}}>c_{C}^{\textit{SN}}\) and \(c_{I}^{\textit{PN}}>c_{I}^{\textit{SN}}\) were checked numerically.

  1. (b)

    \(a_{C}^{\textit{PN}}>a_{I}^{\textit{PN}}>a_{I}^{\textit{SN}}=a_{C}^{\textit{SN}}\): we already proved that \(a_{C}^{\textit{PN}}>a_{I}^{\textit{PN}}\).

    $$\begin{aligned} K_{1}K_{4}\frac{a_{I}^{\textit{PN}}-a_{I}^{\textit{SN}}}{n}=k\omega W\left( X+Y-\omega \right) \left( Y+kW+XY+1\right) +X^{2}YkW>0. \end{aligned}$$
  2. (c)


    $$\begin{aligned} K_{1}K_{4}\frac{m_{C}^{\textit{PN}}-m_{C}^{\textit{SN}}}{kn^{2}p}= & {} -YkW\omega \left( \omega \left( nq-1\right) \left( n^{2}p^{2}-1\right) +n^{2}p^{2}\right) \\&-XY\left( \omega +1\right) \left( W-\omega \right) <0 \end{aligned}$$

    \(m_{I}^{\textit{PN}}\gtrless m_{I}^{\textit{SN}}\):

    $$\begin{aligned} K_{1}K_{4}\frac{m_{I}^{\textit{PN}}-m_{I}^{\textit{SN}}}{kn\omega ^{2}}= & {} kW\left( n^{2}p^{2}\omega \left( np-1\right) \left( np+1\right) \left( nq-1\right) \right) \\&-\left( n^{2}p^{2}\omega +1\right) \left( nq\omega +1\right) \left( n^{2}p^{2}+nq-1\right) \\&-\,kW\left( n^{2}p^{2}+nq-1\right) . \end{aligned}$$

    \(m_{I}^{\textit{PN}}>m_{I}^{\textit{SN}}\) if both these conditions are satisfied:

    $$\begin{aligned} \omega>&\frac{n^{2}p^{2}+nq-1}{n^{2}p^{2}\left( np-1\right) \left( np+1\right) \left( nq-1\right) }>0 \\ \left( \theta \omega -1\right)>&\frac{\left( n^{2}p^{2}\omega +1\right) \left( nq\omega +1\right) \left( n^{2}p^{2}+nq-1\right) }{W\left( n^{2}p^{2}\omega \left( np-1\right) \left( np+1\right) \left( nq-1\right) -\left( n^{2}p^{2}+nq-1\right) \right) }>0. \end{aligned}$$

\(\square \)

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Breton, M., Sbragia, L. Adaptation to Climate Change: Commitment and Timing Issues. Environ Resource Econ 68, 975–995 (2017).

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  • Adaptation
  • Climate change
  • Leadership
  • Mitigation
  • Strategy
  • Timing