Cooperative Action on Greenhouse Gas Emissions and the Distribution of Global Output and damage

Abstract

This paper compares the performance of two allowance allocation rules in an international climate change treaty. I construct a model in which countries differ according to both GDP and an idiosyncratic damage parameter that links global emissions to damage for an individual country. Allowances are allocated to treaty members according to an allocation rule based on a single allocation parameter. The model can be solved analytically to determine upper and lower bounds on this allocation parameter that ensure that a treaty of any given composition is internally and externally stable. I focus on grand coalition treaties. The first treaty examined uses a simple proportional rule in which allocations are set as some fraction of emissions in the non-cooperative equilibrium. The second treaty adds a coverage-contingent element to the allocation rule such that the emissions reduction required of each treaty member is weighted by the fraction of global emissions that treaty members as a whole emit in the non-cooperative equilibrium. I show that the coverage-contingent treaty outperforms the simpler treaty except when all countries have the same emissions intensity in the non-cooperative equilibrium (in which case neither treaty can reduce emissions).

Appendix

1.1 Derivation of (10)

Set $Z_{-i} =Z-z_{i}$ in (9) and cross-multiply to obtain

$$\begin{aligned} z_{i}\left( 1+\delta _{i}y_{i}^{2}\right) =y_{i}- \delta _{i}y_{i}^{2}(Z-z_{i}) \end{aligned}$$

(56)

Collect terms to obtain

$$\begin{aligned} z_{i}=y_{i}-\delta _{i}y_{i}^{2}Z \end{aligned}$$

(57)

and then sum across $i$ to obtain

$$\begin{aligned} Z=Y-Z\sum _{i=1}^{n}{\delta _{i}y_{i}^{2}} =Y-ZS \end{aligned}$$

(58)

which then yields (10).

1.2 Analysis of (41)

Rewrite condition (41) as

$$\begin{aligned} \sum _{i\ne L} {w_{Ci} v_{i}} <v_{L}-w_{CL} v_{L}\quad \hbox {for}\,\,i\in C \end{aligned}$$

(59)

The RHS can be written as

$$\begin{aligned} v_{L}(1-w_{CL})=v_{L}\sum _{i\ne L} {w_{Ci}} \end{aligned}$$

(60)

since $w_{CL} +\sum _{i\ne L} {w_{Ci}} =1$. Thus, the condition (41) becomes

$$\begin{aligned} \sum _{i\ne L} {w_{iC} v_{i}} <\sum _{i\ne L} {w_{iC} v_{L}} \end{aligned}$$

(61)

This can never hold since $v_{L}=\mathop {\hbox {min}}\nolimits _{i\in C} \{v_{i}\}$ and $v_{i}\ge 0 \,\forall i$.

1.3 Proof of Proposition 2

We know from (20) that demand for allowances by member country $i$ is

$$\begin{aligned} z_{i}^{C}(\beta )=\left( {\frac{y_{i}}{Y_{C}}}\right) Z_{C}(\beta ) \quad \forall i\in C \end{aligned}$$

(62)

where

$$\begin{aligned} Z_{C}(\beta )=(1-\beta )\left( {Y_{C}-\frac{YS_{C}}{1+S}}\right) \end{aligned}$$

(63)

The emissions allowance for country $i$ under the EPR treaty is

$$\begin{aligned} a_{i}(\beta )=(1-\beta )y_{i}\left( {1-\frac{v_{i}Y}{1+S}}\right) \end{aligned}$$

(64)

where $v_{i}=\delta _{i}y_{i}$. Thus, allowance sales by country $i$ are

$$\begin{aligned} s_{i}(\beta )\equiv a_{i}(\beta )-z_{i}^{C}(\alpha )= (\tilde{v}_{C}-v_{i})(1-\beta )\frac{y_{i}Y}{1+S} \end{aligned}$$

(65)

where

$$\begin{aligned} \tilde{v}_{C}=\frac{S_{C}}{Y_{C}}=\sum _{i\in C} {w_{Ci} v_{i}} \end{aligned}$$

(66)

It follows that $s_{i}(\beta )>0$ if and only if $v_{i}<\tilde{v}_{C}$, and $s_{i}(\beta )<0$ if and only if $v_{i}>\tilde{v}_{C}$.

1.4 Sketch Proof of Proposition 3

Setting $\pi _{i}^{C}(\beta )=0$ and solving for $\beta$ yields two roots for each value of $v_{i}$ as a function of $Y_{C}$ and $S_{C}$. As noted in the text, these solutions are too complicated to report usefully here but they are easily generated using Maple (Version 15). The solutions are much simpler when $Y_{C}=Y$ and $S_{C}=S$, and reduce to the roots reported in the text. Note that these two roots are symmetric around a common intercept at $v_{i}=\tilde{v}$.

1.5 Proof of Proposition 4

Setting $Y=Y_{C}$ and $S=S_{C}$ in $\pi _{i}^{C}(\theta )=0$, and solving for $\theta$ using Maple (Version 15), yields the solution for $\bar{{\theta }}_{i}^{GC}$ as a function of $y_{i}$ and $\delta _{i}$ (too complicated to report here). From Proposition 3, let $\bar{{\beta }}_{Li}^{GC}$ denote the solution for $\bar{{\beta }}_{i}^{GC}$ when $v_{i}<\tilde{v}$, and let $\bar{{\beta }}_{Hi}^{GC}$ denote the solution for $\bar{{\beta }}_{i}^{GC}$ when $v_{i}>\tilde{v}$. Setting $\bar{{\theta }}_{i}^{GC} =\bar{{\beta }}_{Li}^{GC}$ and solving for $y_{i}$ yields a single real root, at $y_{i}=0$, for any $\delta _{i}$. Thus, $\bar{{\theta }}_{i}^{GC}$ lies either everywhere above or everywhere below $\bar{{\beta }}_{LI}^{GC}$ for $y_{i}>0$. At $y_{i}=0$, $\partial (\bar{{\theta }}_{i}^{GC} -\bar{{\beta }}_{Li}^{GC})/\partial y_{i}=\delta _{i}Y(1+S)/(1+2S)^{2}>0$ for any $\delta _{i}>0$. Thus, $\bar{{\theta }}_{i}^{GC}$ must lie everywhere above $\bar{{\beta }}_{LI}^{GC}$ for $y_{i}>0$ and $\delta _{i}>0$.

Setting $\bar{{\theta }}_{i}^{GC} =\bar{{\beta }}_{Hi}^{GC}$ and solving for $y_{i}$ also yields a single real root, at $y_{i}=(1+S)/(\delta _{i}Y)$. This is an iso-$v$ contour in $(y_{i},\delta _{i})$ space, along which $v_{i}=v^{*}\equiv (1+S)/Y$. Crucially, along this contour, $z_{i}^{*}=0$; see (12) in the text. If $z_{i}^{*}>0 \,\forall i$ (all countries have positive emissions in the NCE), then $v_{i}<v^{*} \,\forall i$. Thus, if $z_{i}^{*}>0 \,\forall i$ then $\bar{{\theta }}_{i}^{GC}$ lies either everywhere above or everywhere below $\bar{{\beta }}_{Hi}^{GC}$. At $y_{i}=(1+S)/(\delta _{i}Y)$, $\partial (\bar{{\theta }}_{i}^{GC} -\bar{{\beta }}_{Hi}^{GC})/\partial y_{i}$ and $\partial (\bar{{\theta }}_{i}^{GC} -\bar{{\beta }}_{Hi}^{GC})/\partial \delta _{i}$ are both complicated expressions but they are both negative if

$$\begin{aligned} \delta _{i}>-\frac{(1+S)\left[ 1+S+\left( S^{2}+2S+5\right) ^{\frac{1}{2}} \right] }{2Y^{2}} \end{aligned}$$

(67)

This holds for any $\delta _{i}\ge 0$. Thus, $\bar{{\theta }}_{i}^{GC}$ must lie everywhere above $\bar{{\beta }}_{Hi}^{GC}$ for $v_{i}<v^{*}.$

Since $\bar{{\theta }}_{i}^{GC}$ lies everywhere above $\bar{{\beta }}_{LI}^{GC}$ for $y_{i}>0$ and $\delta _{i}>0$, and $\bar{{\theta }}_{i}^{GC}$ lies everywhere above $\bar{{\beta }}_{Hi}^{GC}$ for $v_{i}<v^{*}$, it follows that $\mathop {\hbox {min}}\nolimits _{i\in GC} \{\bar{{\theta }}_{i}^{GC} \}>\mathop {\hbox {min}}\nolimits _{i\in GC} \{\bar{{\beta }}_{i}^{GC}\}$ if all countries have positive emissions in the NCE, except when $v_{i}$ is the same for all countries. In that special case, $\bar{{\theta }}_{i}^{GC} =\bar{{\beta }}_{Li}^{GC} =\bar{{\beta }}_{Hi}^{GC} =0 \,\forall i$.