Equity as a Prerequisite for Stability of Cooperation on Global Public Good Provision

Abstract

Analysing cooperative provision of a global public good such as climate protection, we explore the relationship between equitable burden sharing on the one hand and core stability on the other. To assess the size of the burden which a public good contribution entails for a country, we make use of a specific measure based on Moulin (Econometrica 55:963–977, 1987). In particular, we show that a Pareto optimal allocation which is not in the core can always be blocked by a group of countries with the highest Moulin sacrifices. In this sense, it is the ‘overburdening’ and thus ‘unfair’ treatment of some countries that provides the reason for core instability. By contrast, a Pareto optimal allocation is in the core if the public good contributions are fairly equally distributed according to their Moulin sacrifices. The potential implications of our theoretical analysis for global climate policy are also discussed.

Appendix

1.1 Proof of the Lemma

(i) If: From inequality (5) in the main text, we get $d:=\sum _{i\in K} {(y_i -h_i^A (\tilde{G}))-C(\tilde{G})>0}$. Thus, the allocation $B(K)$ defined by $G^{B(K)}=\tilde{G}$ and $x_i^{B(K)} =h_i^{B(K)} (\tilde{G})+\frac{d}{k}$ for all $i\in K$ is feasible for subgroup$K$, since it fulfils condition (1) of the main text. Clearly, $B(K)$ blocks the initial allocation $A$, since all countries in subgroup $K$ are strictly better off in $B(K)$ than in allocation $A$. Because public good supply $G^{B(K)}$ in a blocking allocation being constructed in this way can be chosen as any $\tilde{G}$ which satisfies (5), this also shows the last part of the Lemma.

(ii) Only if: If coalition $K$ can block allocation $A$ by choosing the allocation $B(K)$ with public good supply $G^{B(K)}$, then condition (2) directly gives $h_i^A (G^{B(K)})<x_i^{B(K)}$ for all $i\in K$. From equation (1) in the main text it thus follows that condition (5) is fulfilled for $\tilde{G}=G^{B(K)}$. But if this is the case for some $\tilde{G}$ there even is a public good supply level $\tilde{G}<G^{A}$ for which inequality (5) also holds. To show this we start from the Samuelson condition for the Pareto-optimal allocation $A$ which reads as $\sum _{i=1}^n {\frac{\partial h_i^A }{\partial G}} (G^{A})+{C}'(G^{A})=0$. (since $mrs_i (h_i^A (G^{A}),G^{A})=-\frac{\partial h_i^A }{\partial G}(G^{A})$, this formulation of the Samuelson condition is equivalent to that stated at the beginning of Sect. 2.2). As $\frac{\partial h_i^A }{\partial G}<0$ especially for all $i\notin K$, then $\frac{\partial Y_K^A }{\partial G}(G^{A})=\sum _{i\in K} {\frac{\partial h_i^A }{\partial G}} (G^{A})+{C}'(G^{A})>0$. Convexity of indifference curves and $_{ }{C}''(G)\ge 0$ furthermore imply that $\frac{\partial Y_K^A }{\partial G}(G)>0$ holds also for all $G\ge G^{A}$, and thus that $Y_K^A (G)>Y_K^A (G^{A})$ for all $G>G^{A}$. Combined with continuity of the function $Y_K^A (G)$ this entails that condition (5) can be satisfied for some $\tilde{G}<G^{A}$ as soon as it can be satisfied at all.

1.2 Proof of Proposition 1

If some coalition $K$ can block a given allocation $A$, the Lemma gives that $G^{B(K)}<G^{A}$ can be assumed for the public good supply $G^{B(K)}$ in the blocking allocation. We can then show that the coalition defined by ${K}':=\left\{ {i\in N:G^{B(K)}\ge \bar{{G}}_i^A } \right\}$ is also able to block allocation $A$. As indifference curves are downward sloping, we clearly have ${K}'=\left\{ {i\in N:h_i^A (G^{B(K)})\le y_i } \right\}$, which implies the following relations:

$$\begin{aligned} \sum \limits _{i\in {K}'} (y_i \!-\!h_i^A (G^{B(K)}))\!\ge \! \sum \limits _{i\in {K}'\cap K} (y_i \!-\!h_i^A (G^{B(K)}))\!\ge \! \sum \limits _{i\in K} (y_i \!-\!h_i^A (G^{B(K)}))\!>\!C(G^{B(K)}).\quad \end{aligned}$$

(8)

The first inequality in (8) is satisfied, since $y_i -h_i^A (G^{B(K)})\ge 0$ for all $i\in {K}'$ follows from the definition of ${K}'$, and since, clearly, ${K}'\cap K\subseteq {K}'$. The second inequality is obtained because, for all countries $i$ that are in $K$ but not in ${K}'$, we have $y_i -h_i^A (G^{B(K)})<0$. The third inequality is implied by the ‘only-if’ part of the Lemma. The ‘if-part’ of the Lemma then shows that the coalition ${K}'$ is also able to block allocation $A$. Finally, we define , implying that . Thus, gives the requested threshold level for the Moulin sacrifices.

1.3 Proof of Proposition 2

We first show that, for any subgroup $K$, there exists an $\varepsilon (K)>0$ such that all Pareto-optimal allocations $A$ for which $s_i^A \in \left[ {s^{E}-\varepsilon (K),s^{E}+\varepsilon (K)} \right]$ holds for all $i=1,\ldots ,n$ cannot be blocked by coalition $K$. Otherwise, there would exist a sequence $(A_j )_{j\in {\mathbb {N}}}$ of Pareto-optimal allocations (and corresponding sequences of Moulin sacrifices $(s_i^{A_j } )_{j\in {\mathbb {N}}}$ and utility levels $(u_i^{A_j } )_{j\in {\mathbb {N}}}$ with $\lim \limits _{j\rightarrow \infty } s_i^{A_j } =s^{E}$ and $\lim \limits _{j\rightarrow \infty } u_i^{A_j } =u^{E}$ for each $i=1,\ldots ,n)$ such that, for all $j\in {\mathbb {N}}$, coalition $K$ could block allocation $A_j =(x_j^{A_j } )_{j\in {\mathbb {N}}}$. Then, the Lemma implies that $\hat{{Y}}_K^{A_j } <Y_K$ for all $j\in {N}$. As $\lim \limits _{j\rightarrow \infty } u_i^{A_j } =u^{E}$, this gives $\hat{{Y}}_K^E =\lim \limits _{j\rightarrow \infty } \hat{{Y}}_K^{A_j } <Y_K$, which—again according to the Lemma—means that coalition $K$ could attain the same utility levels for all its members by standing alone and choosing a blocking allocation with some public good supply $G^{B(K)}$. Since, trivially, $G^{B(K)}>G^{E}-s^{E}$, the allocation in which the coalition $K$ chooses $B(K)$ and all countries $i\notin K$ enjoy private consumption $y_i$ is feasible and—as $h_i^A (G^{B(K)})<y_i$ for all $i=1,\ldots ,n$—would entail a Pareto improvement over $E$. This, however, contradicts the Pareto optimality of $E$. Thus, a sequence $(A_j )_{j\in {\mathbb {N}}}$ of blockable allocations converging to $E$ cannot exist, implying that there is a critical value $\varepsilon (K)>0$ with the required property for any coalition $K$. Taking the minimum of all $\varepsilon (K)$ over the finitely many subgroups of $N$ then completes the proof.

1.4 Proof of Proposition 3

Given some coalition $K$ let $\underline{s}(K):=s^{E}+(\bar{{G}}^{E}-\bar{{G}}^{E(K)})$ be the threshold level for Moulin sacrifices, where $\bar{{G}}^{E}$ and $\bar{{G}}^{E(K)}$ are the egalitarian-equivalent public good supply levels for the EMS $E$ of all countries and for the standalone equal Moulin sacrifice solution $E(K)$ of group $K$, respectively. Now consider any allocation $A$ for which $s_i^A >\underline{s}(K)$ holds for each country $i\in K$. Then we have

$$\begin{aligned} \bar{{G}}_i^A= & {} G^{A}-s_i^A <G^{A}-(s^{E}+(\bar{{G}}^{E}-\bar{{G}}^{E(K)}))=G^{A}-(({G}^{E}-\bar{{G}}^{E})\nonumber \\&+\,\,(\bar{{G}}^{E}-\bar{{G}}^{E(K)}))=\bar{{G}}^{E(K)}. \end{aligned}$$

(9)

In (9) the first and the second equality signs follow from the definition of the Moulin sacrifices, the inequality sign from the assumption $s_i^A >\underline{s}(K)$ and the definition of $\underline{s}(K)$, and the last equality sign from $G^{A}=G^{E}$ because with CEP preferences public good supply is the same in all Pareto optimal allocation. From $\bar{{G}}_i^A <\bar{{G}}^{E(K)}$, as established by (9), it follows that $u_i (x_i^{E(K)} ,G^{E(K)})=u_i (y_i ,\bar{{G}}^{E(K)})>u_i (y_i ,\bar{{G}}^{A})=u_i (x_i^A ,G^{A})$ for every $i\in K$ so that all members of coalition $K$ attain a higher utility in their standalone allocation $E(K)$ than in the allocation $A$. Therefore, coalition $K$ is able to block allocation $A$.

Finally, $\bar{{G}}^{E(K)}<\bar{{G}}^{E}$ and thus the inequality $\underline{s}^{E}>s^{E}$ hold since otherwise the feasible allocation, in which private consumption is $x_i^{E(K)}$ for all $i\in K$ and $y_i$ for all $i\notin K$ and public good supply is $G^{E(K)}$, would Pareto dominate allocation $E$, which is a contradiction to the Pareto optimality of $E$.

1.5 Proof of Proposition 4

We rank countries in descending order of their Moulin sacrifices, i.e. $s_1^A \ge \ldots \ge s_n^A$. Then, initial endowment is shifted from some country $j$ to another country $l>j$ such that the Moulin sacrifice of country $j$ decreases and that of country $l$ increases while the ranking of the sacrifice levels is preserved. Given the technological assumption T1 the allocation $A$ remains feasible after such a redistribution of initial endowment since all private consumption levels are kept constant. For any $k=1,\ldots ,n$, now let total income of subgroup $K(k):=\left\{ {1,\ldots ,k} \right\}$ be denoted by $Y_k$ before and by $\tilde{Y}_k$ after the transfer. Obviously, $\tilde{Y}_k =Y_k$ holds for all $k<j$ and all $k\ge l$, while $\tilde{Y}_k <Y_k$ for all $k$ with $j\le k<l$. The Lemma then implies that no coalition $K(k)$ is able to block the original allocation $A$ after the change if no coalition $K(k)$ could do so under the initial income distribution. Proposition 1, however, says that some of these coalitions $K(k)$ (with the relatively highest Moulin sacrifices) should have been able to block $A$ if this allocation can be blocked at all. This shows that allocation $A$ remains in the core after the income redistribution.