2.1. Two-step procedure

To achieve a (world) social optimum with carbon emissions,Footnote ⁸ one has to consider two distinct maximization problems. The first is to determine optimum carbon abatement applying standard utilitarian principles. Efficiency is achieved by equalizing marginal cost and marginal benefits of abatement. The 2^°C target of climate policy can be interpreted as a (broadly accepted) solution to this maximization problem. Based on the temperature target, science results have to determine the associated world carbon budget available in the future; see Meinshausen et al. (Reference Meinshausen, Meinshausen and Hare2009) for a calculation.

The second maximization problem consists in finding an agreeable treaty, realizing the potential net benefit of climate policies. Such an agreement is by no means automatically reached, even when the aggregate benefit is very large. Quite to the contrary, when assessing the previous conferences of the parties of the UNFCCC, negotiation costs have even proved to be enormous. A binding constraint is the limited time available for further negotiations. Hence, the second optimization problem is to obtain a generally acceptable climate deal, given the heterogenous interests of countries and the transaction costs of climate negotiations. The present paper takes the target world carbon budget as given and presents conditions under which a policy to achieve such a budget appear to be more easily acceptable by all the countries.

Formally, I study the process of finding an allocation of emission budgets Z _j for countries j = 1, 2, …, N, i.e., a series

(1)$$Z_{1}\comma \; Z_{2}\comma \; Z_{3}\comma \; \ldots Z_{N}.$$

In order to be a part of the set of feasible solutions, an allocation (1) has to obey several restrictions. Denoting world target carbon budget (which is compatible with the 2^°C target) by $\bar{Z}$, net benefits of an international climate agreement by B, and negotiation costs by Q, the restrictions read

(2)$$\sum\limits^N_{j=1} Z_j \le \bar{Z}$$

(3)$$B-Q\gt0$$

(4)$$B_j-Q_j\gt0\comma \; \forall j.$$

While (2) is the world emission quantity constraint, (3) and (4) express that net benefits must exceed costs of negotiation both at the global and the country level. As I did not spell out preferences, net benefits in (4) may also include altruism towards low-developed countriesFootnote ⁹ and future generations; (4) corresponds to the assumption of ‘international paretianism’ of Posner and Weisbach (Reference Posner and Weisbach2012), but additionally includes negotiation costs. In the following I focus on condition (3) which, with given B, can only be met when Q does not grow too large. Accordingly, the next section proposes a theory of negotiation costs identifying possible ways to reduce Q so that (3) can be satisfied with given B. The following sections will deal with proposals likely to satisfy (3) and (4).

To study the issue in a systematic way, I need to make assumptions about the mechanics of the negotiation process. Notably, I have to specify negotiation technology and associated cost, which I present as a novel approach in the following subsection.

2.2. Negotiation costs

I argue that negotiation costs Q are transaction costs and assume them to depend on the number of negotiating parties, the number and complexity of the negotiation items, the heterogeneity of interests between the parties, and the negotiation setting which determines how parties interact during negotiations. Often-used settings at international climate negotiations are plenary sessions, specialized sessions, country group meetings and bilateral talks. Experience of the climate negotiations shows that many decisions taken in specialized working groups or plenary sessions are solidly prepared in bilateral talks. Accordingly, the following approach to modeling negotiation costs uses the costs of bilateral talks as an element.

I first look at an extreme case, i.e., the maximum negotiation cost to obtain an agreement, and then discuss various ways to reduce the costs. If negotiations of a specific item like abatement targets require going through all possible bilateral meetings with N different parties, a total of (N ² − N)/2 meetings are necessary, which shows the disproportional impact of the number of parties. Of course, a realistic setting helps to omit part of these meetings; I will discuss below which items are most likely to be successful in this respect. To obtain costs, the number of meetings has to be multiplied by average meeting costs, which depend positively on the number and the complexity of the negotiation items and the heterogeneity of parties' interests.

Theory has to focus on the fact that it usually takes several negotiation rounds before convergence between the involved parties is obtained. I label the different negotiation rounds by t = 1, 2, 3, … and express negotiation costs in t by $M\lpar t\rpar =\bar{M}\cdot V\lpar t\rpar $ where $\bar{M}$ is a negotiation-specific constant and V a convergence factor, which increases between the rounds but converges to a steady state; an agreement requires V(t + 1) = V(t).Footnote ¹⁰ Typically, a climate policy negotiation document includes different sub-items which are sequentially agreed on by the parties. The dynamic process of V depends on the speed by which parties are able to agree on these sub-items. Formally, I assume that V evolves according to

(5)$$V\lpar t+1\rpar =\left(1-\phi\right)V\lpar t\rpar +\chi\left[V\left(t\right)\right]^{\xi}$$

where 0 < ϕ,ξ,χ < 1. Expression (5) allows discussion of the different dynamic elements of negotiation costs. The ‘completion’ parameter ϕ denotes the share of negotiation sub-items which has been agreed on in the previous round. The ‘step’ rate χ determines the increase of V between the negotiation rounds; if ξ = 1 the growth rate of V equals χ − ϕ, i.e., it diminishes fast with low χ, meaning a fast decrease of negotiation complexity. As an example, negotiations concerning technical issues like carbon capture and sequestration may decrease rapidly in complexity once the technicalities are well understood. The marginal impact of χ[V(t)]^ξ on V(t + 1) decreases with rising V which results in convergence. The convergence term ξ is low when additional negotiation rounds, increasing V, allow for rapid convergence; when it is high, convergence is slow. An example with little convergence and hence high ξ has been the negotiation on technology transfers under the Bali action plan where country positions remained controversial for several years. The growth rate of V is given by χ[V(t)]^ξ−1 − ϕ so that V converges to V ^* = (χ/ϕ)^1/(1−ξ) and aggregate negotiation costs Q for a specific negotiation item are given by

(6)$$Q={\bar{M}\lpar N^{2}-N\rpar \over{2}}\left({\chi\over\phi}\right)^{1/\lpar 1-\xi\rpar }.$$

It becomes clear from (6) that high specific cost $\bar{M}$, a high step rate χ and a large convergence parameter ξ entail high negotiation cost Q, while a large completion share ϕ decreases cost.

Expression (6) allows us to analyze different negotiation items like abatements targets or burden-sharing rules with respect to their impact on Q in (6). The number of parties N is given at the United Nations but the (quadratic) term in (6) is based on the assumed complete bilateralism. Any items reducing the number of necessary bilateral meetings thus decrease the impact of N on cost Q. Specifically, if an item is largely acceptable, countries may decide to negotiate in groups (like the G77), which reduces the number of necessary talks substantially. We then have a theory on bilateral talks between groups. Besides the obvious effects of $\bar{M}$, χ and ϕ, it is the convergence parameter ξ which is of special interest, because it has a disproportional impact on Q. A very complex item can cause slow convergence, i.e., a large number of additional negotiation rounds. This happens with a value of ξ close to unity, increasing total negotiation cost Q to a very large amount.

2.3. Negotiation items

I now look at different possible negotiation items and burden-sharing proposals at international climate negotiations with regard to the crucial model parameters N and ξ in (6).

• • A pledge and review procedure for country abatement targets reveals countries' willingness to participate but has produced very unequal results between the countries. It is unclear to many, specifically to the non-pledging countries, how the offers compare between the countries, especially in terms of economic costs of climate policies. Recent climate conferences have proved that countries do not assess that the process generates a fair burden sharing; it has not produced sufficient convergence so far, indicating high ξ. Hence, the procedure seems to imply unfavorable values for N and ξ and thus a high Q.

• • A second strategy is to include all the aspects determining the consequences of the pledges, i.e., to negotiate about economic cost, trade positions, money transfers, technology development and more. This increases the number of items drastically, which is not a very promising approach to bring about more favorable values for N and ξ.

• • The third strategy is on a very different level, proposing equity principles for the allocation of the carbon budget to countries which are generally accepted and considered to be fair. According to (6), the lower the number and the complexity of the negotiations items, the lower the values of N and ξ become, which improves the prospects of finding an agreement. Consequently, the paper proposes a burden-sharing rule based on a very low number of equity principles. This is according to the requirement for low values of N and ξ, i.e., low negotiation cost.

There are more possible variants but the paper moves on to the next topics for the sake of brevity. To conclude, the aim should be to propose a solution for burden sharing which is broadly considered as fair, clear and simple. This appears most promising when restrictions (3) and (4) have to be satisfied. Equity principles are a natural point of departure for such a reasoning. That in the end each and every country is willing to agree to such a climate deal might also imply some punishment for non-signatory countries. Specifically, if the condition (4) cannot be fulfilled without sanctioning non-signatories, trade sanctions could be put forward as a possible compliance mechanism, like under the Montreal Protocol. Trade sanctions may have undesirable effects but be compatible with free trade agreements in the case of border adjustments with obligations to buy allowances. The paper does not elaborate on this point but concentrates on equity principles as a means to reduce negotiation costs Q in the following.

4.1. Using the principles

There are many equity principles and different ways to apply them in climate policy proposals. The purpose of this subsection is to select some of the most accepted principles according to the literature (see, e.g., Lange et al., Reference Lange, Vogt and Ziegler2007; Mattoo and Subramanian, Reference Mattoo and Subramanian2010). In the next subsection I will complement them with a dynamic perspective which is crucial for the long-run sustainability aspects. The main guideline of the following consideration is to find a combination of equity principles which yields a clear and simple allocation rule for international carbon budgets. This is done in order to find a solution to the problem of (too) high negotiation cost, as analyzed above. I take that by the agreement on the Durban platform all the countries are included in the analysis. In addition, by the Cancun agreements, I assume that the 2^°C target is confirmed. One can then adopt from natural science results what the remaining world carbon budget will be. Naturally, my calculation method could also be used for alternative carbon budgets and/or temperature targets.

The Ability to Pay Principle is included to derive the budgets as a measure of purchasing power, mainly for carbon-efficient technologies. The Cost Sharing Principle is employed to reflect the distribution of abatement costs (not to guarantee an efficient policy, this is done by allowing for permit trade). The Desert Principle is used for the provision of carbon-saving technical progress. The Polluter Pays Principle is included in the sense that past emissions are deducted from the overall carbon budget, possibly with a discount, depending on the valuation of historic responsibility. In the long-run steady state, one can assume that every individual will obtain an equal share of atmospheric resources, but this has no impact on my derivation, because I look at the transition of the economy towards sustainable development.

4.2. Economic dynamics

An important but so far neglected issue for an equitable solution in climate policy is the impact of economic dynamics, in particular technological development. When applying equity principles to climate policies one should compare like with like. Specifically, carbon emissions should be evaluated with respect to the technical opportunities available at the time of emission.

Regarding carbon saving technologies, recent changes have indeed been remarkable. According to World Bank (2012) data, the average annual growth rate of world CO₂ emission intensity (global CO₂ emissions per constant $ of world GDP) for the period 1980–2008 amounts to −1.42 per cent. This means that, on average, the necessity to emit a ton of CO₂ to produce a unit of output decreases by more than 1 per cent each year, which is respectable given the long time horizon of climate policy. The average annual growth rate of world CO₂ emission intensity per capita even amounts to −2.9 per cent.Footnote ¹⁴ World emission intensity and the respective growth rates are depicted in figure 1. Relating this finding to a fair burden allocation suggests that later developing countries have the advantage to have access to technologies with higher carbon efficiency, due to the diffusion of these technologies on world markets.

Fig. 1.

World emission intensity 1980–2008

source: World Bank (2012).

4.3. Fairness index

The different principles are used to build a ‘fairness’ index F, which is a weighting index for fairness, used as the main ingredient to determine fair carbon budgets; a higher F will yield a higher budget in the following. F depends on four variables: the (inverse of the) Ability to Pay A, the abatement cost parameter C, the (individual or country) technology contribution x, and the general technology development X. All variables are measured for an individual i at the beginning of the considered time period. An important methodological issue is how to combine these four factors mathematically, which ultimately determines their weight in the index. A broadly used technique is the weighted average, which relies on the (simple) arithmetic weighted average of the (normalized) variables. When the weights are the same for all the variables we get the so-called ‘equally weighted averages’. The procedure is simple but the linearity implies constant substitutability among the different criteria, which is a strong assumption for international climate policy. It implies that a country can compensate (even extreme) deviations from the mean in one variable in exchange for other variables at a constant ratio. A more suitable procedure is to use the multiplicative form between the variables and define the weights in the exponents. In the present case, this leads to:

(7)$$F_{i}=A_{i}^{\alpha}\cdot C_{i}^{\beta}\cdot x_{i}^{\gamma}\cdot X_{i}^{\delta}$$

where 0 < α,β,γ,δ < 1 determine how the index F increases when a variable on the right-hand side grows. Because the exponents are below unity, the index grows less than proportional with the single variables. For the following application, we measure each of the core variables by the best available indicator, knowing well that a certain degree of roughness is unavoidable with the procedure (in fact, with any procedure). In concrete terms, we capture A by the inverse of income Y per capita (L), the mitigation cost burden by emissions E per capita L, the contribution to technology development by output Y per emission E, and the context of general technology development by emissions E per capita L. This last measure needs further explanation. To take initial emissions per capita as an indicator for future carbon technology development (which for obvious reasons is not directly measurable) relies on the assumption that the function of carbon efficiency development is monotonous. Put differently, the measure is meaningful when we assume that technical efficiency of carbon use is improving steadily, as has been the case in the past (see the previous subsection). Then, a person/country with high pollution in the beginning and a decreasing carbon profile is different from a person/country with low pollution in the beginning and rising emissions. With regard to carbon use, the first person/country has unambiguously less technical opportunities than the second. The measurement of the variables is represented in table 1.

Table 1.

Measuring the variables

Using this specification we get

(8)$$\eqalign{F_{i}&=\left({L_i \over Y_i}\right)^{\alpha}\left({E_i \over L_i}\right)^{\beta+\delta}\left({Y_i \over E_i}\right)^{\gamma} \cr & \quad=L_{i}^{\alpha-\beta-\delta}E_{i}^{\beta+\delta-\gamma}Y_{i}^{\gamma-\alpha}.}$$

In a non-growing (static) economy, the technology development parameter is zero, i.e., δ = 0. Then, applying the equal weight method to the parameters in the exponent, we set α = β= γ. Interestingly, this yields F _i = 1, which is exactly the egalitarian solution with equal weights for every individual, like in BASIC (2011). But note that the result is not using equal carbon space as something like a first principle; the outcome emerges due to the exactly balancing effects of three (real) first principles; namely ability to pay, abatement costs and technology contribution.

However, I argue that the considered economies are inherently dynamic, mainly driven by technical progress. Hence, taking δ > 0 and applying the equal weight rule to all the four variables, we have α = β= γ =δ in order to obtain

(9)$$F_i=\left(E_i\over L_i \right)^\alpha.$$

This expression has many interesting aspects. It reduces the highly complex problem of burden sharing in international climate policy to one single variable (E _i/L _i) and a unique parameter (α, with α = β= γ =δ). It offers a close link to natural sciences and, moreover, an easy calculation method for the emission budgets. Importantly, with the link to (initial) emissions it opens the door to (possible) political acceptance, much like the grandfathering rules used with past carbon policy, without using political constraints as first principle. The derived relationship is nonlinear, so that the impact of emissions on the index fades with increasing emissions. Nonlinearity is even high, because, with the obvious restriction α + β+ γ +δ = 1, we have α = 0.25.

Following the discussion on negotiation cost, the positive attributes of the simple formula (9) are as follows: (i) it is ethically defensible, (ii) it is likely to cut negotiation cost enormously, and (iii) it has no negative impact on market efficiency when carbon trading is possible after the first allocation. Note that the simplicity of (9) is due to the used weights which are equal. If the assumption is disputed, an increase in negotiation costs is the inevitable consequence.

4.4. Budget calculation

Individual allocation

Each individual i receives an emission budget B _i according to the individual fairness index F _i and the total amount of entitlements Z available, according to

(10)$$B_i={F_i \over \Sigma_{i=1}^{L}F_i}. Z.$$

Country allocation

Let us define the population share of country j as m _j(t) = L _j(t)/L(t) and use N for the number of countries. Each country receives an emission budget Z _j according to the weighting index F _j (reflecting the fair index for country j), the country share m _j, and the total amount of entitlements Z available in total, so that

(11)$$Z_j={m_{j}.F_{j} \over \Sigma_{j=1}^{N} m_{j}. {F_j}}.Z.$$

4.5. Historic responsibility

So far we have not distinguished between different time periods. The whole period to consider starts with historic responsibility and is assumed to end in 2050, because current budget calculations provided by natural sciences end at that point in time. Because of natural decay of greenhouse gases and the absence of legal obligations in the past we might value past carbon use differently from future use. For practical reasons, we will distinguish between a first period, starting with the date of historic responsibility up to the point where the most recent statistics are available, and a second period, from that point in time until 2050. For the budget calculation, we assume the countries to receive entitlements according to the fairness index as derived above but now posit that actually used emissions in the first period may be discounted with a factor 0 < θ< 1. In the concrete calculations, we will compare carbon budgets for the responsibility values θ = 1.0; 0.8; 0.5; and 0.

5.1. Assumptions

To provide concrete information about the derived fairness equation, this section calculates country carbon budgets, using the fairness index F and a specific set of assumptions. Of course, the general concept equally holds when different parameter specifications and time periods are more politically acceptable. The assumptions for the calculations here are:

• • Start of (historic) responsibility: 1990

• • First (‘responsibility’) period; 1990–2008 (1990 is the starting date of international climate policy; for 2008 the last international data on carbon use are available)

• • Second (‘carbon budget’) period: 2008–2050 (feasible world carbon budgets were calculated up to 2050)

• • Total carbon budget Z for 1990–2050: (i) ‘Variant 50 per cent’, i.e., carbon emissions corresponding to a probability of 50 per cent of exceeding 2^°C, and (ii) ‘Variant 33 per cent’, i.e., the equivalent for a probability of 33 per cent, based on Meinshausen et al. (Reference Meinshausen, Meinshausen and Hare2009) who cover 2000–2050, plus carbon emissions 1990–2000, yielding 1,640 gigatons in total for ‘Variant 50 per cent’ and 1,360 gigatons for ‘Variant 33 per cent’.

• • Calculation of E _i/L _i: per capita carbon emission data for 1990 as given by World Bank (2012).

• • α = 0.25 (=β = γ= δ)

• • Population size: 1990 values

• • Valuation of emissions in the ‘responsibility’ period: scenarios for different levels of θ ranging from θ = 1 (full responsibility) to θ → 0 (no responsibility).

5.2. Budgets

Tables 2–4 show the carbon budgets for 2008–2050 for selected countries and four levels of historic responsibility, given by different values of θ. The tables contain two sections, with the ‘Variant 50 per cent’ (1,640 gigatons in total) given on the left and the lower ‘Variant 33 per cent’ (1,360 gigatons) on the right. Table 2 contains the total budget, table 3 the budget per capita (using 2008 population values), and table 4 the average budget per capita per year.

Table 2.

Total budget (period 2008–2050 in gigatons)

Table 3.

Budget per capita (period 2008–2050 in tons)

Table 4.

Average budget per capita per year (period 2008–2050 in tons)

By inspection of the tables it becomes clear that the shares (specifically the budgets per capita) are not very unequal between the different countries. Late-developing countries and high past polluters receive relatively lower budgets. Less-developed economies have the possibility to increase per capita emissions compared to current emissions, at least in a first time period. The common issue for higher developed countries is that they have to cut emissions compared to previous levels, although in a differentiated manner.

As can be seen from the tables, historic responsibility matters especially for the United States, but is also an issue for other countries like Russia or Japan. The outcome suggests that historic responsibility is likely to be an important separate negotiation item. Otherwise, results are quite robust, also with respect to other model parameters.

Note that the values in table 4 are average emission values for the whole period and not final values for 2050, which makes a big difference. The time profile is not predetermined by these calculations.