Unspoken Ethical Issues in the Climate Affair: Insights from a Theoretical Analysis of Negotiation Mandates

Abstract

Taking climate change as an example, this paper provides new insights on the optimal provision of a long-term public good within and across generations. We write the Bowen–Lindhal–Samuelson (BLS) conditions for the optimal provision of the public good in a world divided into N countries, with two periods, present and future, and we simultaneously determine the optimal response in the first and second periods for a given rate of pure time preference. However, the Negishi weights at second period cannot be determined unambiguously, even under a “no redistribution constraint” within each generation, because they depend on non-observable future incomes; and thus on the answers to two often-overlooked ethical questions: (i) Do rich countries agree on deals which recognize that developing countries may catch up with developed countries in the long run, or do they use their negotiating powers to preserve the current balance of power? And (ii) does each country consider only the welfare of its own future citizens (dynastic solidarity) or does it extend its concern to all future human beings (universal solidarity)? Answers to (i) and (ii)—critical in the debate about how to correct the market failures causing global warming—define four sets of Negishi weights and intertemporal welfare functions, which we interpret as four mandates that countries could give to the Chair of an international negotiation on climate change to find an optimal solution. We find that in all mandates, public good provision expenditures are decreasing functions of income at first period. But each mandate leads to a different allocation of expenditures at second period and to different optimal levels of public good provision at both first and second periods. Finally, we show that only one of these four mandates defines a space for viable compromises.

The authors are grateful to François Bourguignon, Graciela Chichilnisky, Sylviane Gastaldo, Roger Guesnerie, Cédric Philibert, Gilles Rotillon, Zmarak Shalizi, Tarik Tazdait, David Wheeler and anonymous referees for very constructive comments on earlier versions of this manuscript. The authors would also like to thank the participants of workshops and seminars in Kyoto, Bilbao, La Réunion, Washington D.C., and Paris for helpful discussion.

Reprinted with kind permission from the Authors: Originally published in Economic Theory, Volume 49, Number 2, February 2012.

Appendices

Appendix 1: Model Resolution

We solve a general version of the planner’s problem (‎26)—in which coefficients χ _i summarize both S and C mandates (27)—under constraints (‎3), (‎ 4), (‎5), and (‎6).

$$ {\text{Max}}_{{\{ a_{i} ,a_{i}^{f} \} }} \sum\limits_{i} {\alpha_{i} l_{i} U_{i} \left( {y_{i} {-}a_{i} } \right)} + \varphi \sum\limits_{i} {\chi_{i} l_{i}^{f} U_{i}^{f} \left( {y_{i}^{f} {-}a_{i}^{f} {-}d_{i} \left( {x + x^{f} } \right),d_{j \ne i} } \right)} $$

(26)

With

$$ \chi_{i} = \left\{ {\begin{array}{*{20}l} {\alpha_{i} = \frac{\alpha }{{{\text{U}}_{\text{i}}^{\prime } \left( {y_{i} } \right)}}\;{\text{in}}\;{\text{S}}\;{\text{mandates}},{\text{with}}\;\alpha = \left( {\sum\limits_{i} {\frac{{l_{i} }}{{{\text{U}}_{1}^{\prime } \left( {y_{i} } \right)}}} } \right)^{ - 1} } \hfill \\ {\beta_{i} = \frac{\beta }{{{\text{U}}_{\text{i}}^{\prime } \left( {y_{i} } \right)}}\;{\text{in}}\;{\text{C}}\;{\text{mandates}},{\text{with}}\;\beta = \left( {\sum\limits_{i} {\frac{{l_{i}^{f} }}{{{\text{U}}_{\text{i}}^{{{\text{f}}\prime }} \left( {y_{i}^{f} } \right)}}} } \right)^{ - 1} } \hfill \\ \end{array} } \right. $$

(27)

Let λ, φ μ, l _i ξ _i and φ l ^f _i ψ _i be the Lagrange multipliers attached to constraints (‎3), (‎4), (‎5), and (‎6) respectively. Since (‎5) and (‎6) are inequality conditions, ξ _i and ψ _i are such that:

$$ \left\{ {\begin{array}{*{20}l} {\xi_{i} = 0} \hfill & {\text{if}} \hfill & {a_{i} > 0} \hfill \\ {\xi_{i} > 0} \hfill & {\text{if}} \hfill & {a_{i} = 0} \hfill \\ \end{array} } \right. $$

(28)

$$ \left\{ {\begin{array}{*{20}l} {\psi_{i} = 0} \hfill & {\text{if}} \hfill & {a_{i}^{f} > 0} \hfill \\ {\psi_{i} > 0} \hfill & {\text{if}} \hfill & {a_{i}^{f} = 0} \hfill \\ \end{array} } \right. $$

(29)

With these notations, the Lagrangean of the problem becomes:

$$ \begin{aligned} L = & \sum\limits_{i} {\alpha_{i} l_{i} U_{i} \left( {y_{i} - a_{i} } \right)} + \varphi \sum\limits_{i} {\chi_{i} l_{i}^{f} U_{i}^{f} \left( {y_{i}^{f} - a_{i}^{f} {-}d_{i} \left( {x + x^{f} } \right),d_{j \ne i} } \right)} \\ & + \,\lambda \left[ {\sum\limits_{i} {l_{i} a_{i} - C(x)} } \right] + \varphi \mu \left[ {\sum\limits_{i} {l_{i}^{f} a_{i}^{f} - C^{f} \left( {x^{f} } \right)} } \right] + \sum\limits_{i} {l_{i} \xi_{i} a_{i} } + \varphi \sum\limits_{i} {l_{i}^{f} \psi_{i} a_{i}^{f} } \\ \end{aligned} $$

(30)

The first-order condition with regard to a _i is:

$$ \frac{\partial L}{{\partial a_{i} }} = 0 \Rightarrow \alpha_{i} U_{i}^{\prime } \left( {y_{i} - a_{i} } \right) - \xi_{i} = \lambda $$

(31)

We first demonstrate that (‎31) has a unique solution with strictly positive abatement levels a _i . Let us assume without loss of generality that a ₁ were zero. Since ξ ₁ > 0, α ₁ U ₁′(y ₁ − a ₁) − ξ ₁ = α ₁ U ₁′(y ₁) − ξ ₁ < α ₁ U ₁′(y ₁). If another abatement level, say a ₂, were strictly positive, then α ₂ U ₂′(y ₂ − a ₂) − ξ ₂ = α ₂ U ₂′(y ₂ − a ₂) > α ₂ U ₂′(y ₂) since marginal utilities are strictly decreasing functions. Yet by definition of the Negishi weights, α ₁ U ₁′(y ₁) = α ₂ U ₂′(y ₂). Thus, we would have α ₁ U ₁′(y ₁ − a ₁) − ξ ₁ < α ₂ U ₂′(y ₂ − a ₂) − ξ ₂, contradicting Eq. (31). Thus, if one of the abatement expenditures is zero, all abatement expenditures are zero.

But if all a _i were zero, Eq. (3) and the fact that C(0) = 0 would imply that x = 0, and thus that marginal costs of abatement C′ are zero. Equalization between marginal costs of abatement and marginal damages at optimum (Eq. ‎36) would then imply that all marginal damages d _i ′(x + x ^f) be zero, and thus (via Eq. 33) that the marginal costs of mitigation at second period be zero, and thus that the level of abatement at second period x ^f = 0. But this would contradicts the assumption that no abatement leads to strictly positive marginal damages of climate change. First-period abatement levels a _i are thus all strictly positive, and since marginal utility functions are strictly decreasing, they are uniquely defined.

Similarly, derivation of L with regard to a ^f _i yields:

$$ \frac{\partial L}{{\partial a_{i}^{f} }} = 0 \Rightarrow \chi_{i} U_{i}^{f\prime } \left( {y_{i}^{f} - a_{i}^{f} - d_{i} \left( {x + x^{f} } \right),d_{j \ne i} } \right) - \psi_{i} = \mu $$

(32)

The second part of the argument above can be replicated to demonstrate that at least one of the second-period abatement expenditures a ^f _i is strictly positive. But the first part of the argument above cannot be applied as is, and thus there is no guarantee that all second-period abatement expenditures be strictly positive.

In C mandates, this is because marginal utilities before abatement and after damages $\chi_{i} U_{i}^{f\prime } \left( {y_{i}^{f} {-}d_{i } \left( {x + x^{f} } \right)} \right)$ have no reason a priori to be equal. We only know that marginal utilities of consumption before abatement and before damages $\chi_{i} U_{i}^{f\prime } \left( {y_{i}^{f} } \right)$ are equal (by construction), and thus that the aggregate climate bills a ^f _i  + d _i (and not just the abatement expenditures component) are all strictly positive.

In S mandates, even that weaker property does not hold because even marginal utilities of consumption before abatement and before damages $\chi_{i} U_{i}^{f\prime } \left( {y_{i}^{f} } \right)$ have no reason a priori to be equal across regions.

Derivation of L with regard to x ^f yields:

$$ C^{f\prime } \left( {x^{f} } \right) = \sum\limits_{i} {l_{i}^{f} } \pi_{i} \left[ { - d_{i}^{\prime } \left( {x + x^{f} } \right) + \rho \sum\limits_{j \ne i} {\frac{{\frac{{\partial U_{i}^{f} }}{{\partial d_{j} }}}}{{U_{i}^{f\prime } }}d_{j}^{\prime } \left( {x + x^{f} } \right)} } \right] $$

(33)

With

$$ \pi_{i} = \frac{{\chi_{i} }}{\mu }U_{i}^{f\prime } \left( {y_{i}^{f} - d_{i} \left( {x + x^{f} } \right) - a_{i}^{f} ,d_{j \ne i} } \right) = \frac{{\chi_{i} U_{i}^{f\prime } }}{{\chi_{i} U_{i}^{f\prime } - \psi_{i} }} $$

(34)

Given the assumptions made about marginal damages and marginal abatement costs, (33) has a unique solution. Coefficients π _i are the ratios of χ _i U ^f _i ′ the weighted marginal utility of consumption of country i, and of μ the weighted marginal utility of consumption of the countries that abate at second period (as per Eq. ‎32) (μ can also be interpreted as the shadow price of abatement at second period, expressed in marginal utility terms). When country i contributes to abatement at second period, these two terms are equal and π _i  = 1. When country i does not contribute to abatement—either because it does not grow rapidly enough, or because domestic damages are too high—π _i  > 1.

In other words, when a country has a weighted marginal utility of consumption that is too high relative to the others, not only will it not contribute to abatement, but damages falling on this country will be weighted higher than damages falling on others because they cause higher utility losses at the margin.

When all countries contribute—which, as discussed above, occurs mostly in the C mandates—(‎34) simplifies in (35), which is the standard BLS condition.

$$ C^{f\prime } \left( {x^{f} } \right) = {-}\sum\limits_{i} {l_{i}^{f} d_{i}^{\prime } \left( {x + x^{f} } \right)} $$

(35)

Finally, derivation of L with regard to x yields:

$$ C^{\prime } (x) = \varphi \sum\limits_{i} {l_{i}^{f} } \omega_{i} \left[ { - d_{i}^{\prime } \left( {x + x^{f} } \right) + \rho \sum\limits_{j \ne i} {\frac{{\frac{{\partial U_{i}^{f} }}{{\partial d_{j} }}}}{{U_{i}^{f\prime } }}d_{j}^{\prime } \left( {x + x^{f} } \right)} } \right] $$

(36)

with

$$ \omega_{i} = \frac{{\chi_{i} }}{\lambda }U_{i}^{f\prime } = \frac{{\chi_{i} U_{i}^{f\prime } \left( {y_{i}^{f} - d_{i} \left( {x + x^{f} } \right) - a_{i}^{f} ,d_{j \ne i} } \right)}}{{\alpha_{i} U_{i}^{\prime } \left( {y_{i} - a_{i} } \right)}} $$

(37)

Coefficients ω _i capture the change in weighted marginal utility of consumptions between the first and the second period. Comparing Eqs. (‎33) and (36‎), the term φ ω _i can be interpreted as the region-specific discount rate that applies to climate change damages. In C mandates, this term becomes

$$ \varphi \omega_{i} = \varphi \frac{\beta }{\alpha }\frac{{U_{i}^{\prime } \left( {y_{i} } \right)}}{{U_{i}^{\prime } \left( {y_{i} - a_{i} } \right)}}\frac{{U_{i}^{f\prime } \left( {y_{i}^{f} - d_{i} \left( {x + x^{f} } \right) - a_{i}^{f} ,d_{j \ne i} } \right)}}{{U_{i}^{f\prime } \left( {y_{i}^{f} } \right)}} $$

(38)

When abatement expenditures and damages remain small with regard to income, the last two terms are close to one, and regional discount factors are all equal to $\varphi \frac{\beta }{\alpha }$. Hence coefficient ρ in Eq. (‎23). For example, with logarithmic utility functions, ρ = $\varphi \frac{\beta }{\alpha } = \varphi \frac{1}{{(1 + r)^{N} }}$ where r is the aggregate growth rate of the economy over the first period. In D mandates, on the other hand, regional discount rates become

$$ \varphi \omega_{i} = \varphi \frac{{U_{i}^{f\prime } \left( {y_{i}^{f} - d_{i} \left( {x + x^{f} } \right) - a_{i}^{f} ,d_{j \ne i} } \right)}}{{U_{i}^{\prime } \left( {y_{i} - a_{i} } \right)}} $$

(39)

which vary depending on the regional growth rate between the two periods. For example, with logarithmic utility functions, ρ _i  = $\varphi \frac{1}{{\left( {1 + r_{i} } \right)^{N} }}$ where r _i are the regional growth rates over the first period.

Appendix 2: Data and Modeling Framework of Numerical Experiments

The world is divided in two regions: “North” (N) comprises high-income countries, as per World Bank (2004) definition, and “South” (S) low and middle income countries. The first period is 2000–2049, and the second 2050–2099. Initial income and population data are from World Bank (2004). Average annual economic growth in N is assumed to be 2.5 % between 2000 and 2050, against 3 % in S. World population is assumed to grow by 2 billions in that period of time, all of them in S (Table 4). For simplicity’s sake, we use 2000 (resp. 2050) figures as averages for period 1 (resp. 2).

Table 4 Key assumptions in numerical example

Without abatement, World CO₂ emissions are assumed to be 513 GtCO₂ during the first period, and 688 GtCO₂ during the second, as in the IPCC IS92a scenario (IPCC 1994).

Abatement costs at first and second period are assumed quadratic with respect to total abatement levels. We also assume that mitigating all the emissions in the world economy would cost at the margin $1,500/tC during the first period, and $1,000/tC during the second. After easy manipulations, Eqs. (‎3) and (‎4) become:

$$ x = 1482 \times \sqrt {\frac{{l_{N} a_{N} + l_{S} a_{S} }}{{l_{N} y_{N} + l_{S} y_{S} }}} $$

(40)

$$ x^{f} = 4066 \times \sqrt {\frac{{l_{N}^{f} a_{N}^{f} + l_{S}^{f} a_{S}^{f} }}{{l_{N}^{f} y_{N}^{f} + l_{S}^{f} y_{S}^{f} }}} $$

(41)

Damages are assumed cubic with total emissions.

$$ d_{i} \left( {x + x^{f} } \right) = y_{i}^{f} \theta_{i} \left( {1 - \frac{{x + x^{f} }}{1201}} \right)^{3} $$

(42)

Coefficients θ _i represent the maximum damage—expressed as a share of per capita GDP—that each region may sustain because of climate change. If there were no mitigation at all, damages would be $d_{i} = y_{i}^{f} \theta_{i}$ We use several values for coefficients θ _i to represent various assumptions about the distribution of impacts of climate change across regions—keeping the aggregate maximal damage $l_{S}^{f} y_{S}^{f} \theta_{S} + l_{N}^{f} y_{N}^{f} \theta_{N}$ constant. Finally, all utility functions are assumed logarithmic, and the rate of pure time preference φ is set to 1 %.