Accounting for Inequality Aversion Can Justify the 2◦ C Goal

Impact assessment models are a tool largely used to investigate the benefit of reducing polluting emissions and limiting the anthropogenic mean temperature rise. However, they have been often criticised for suggesting low levels of abatement. Countries and regions, that are generally the actors in these models, are usually depicted as having standard concave utility functions in consumption. This, however, disregards a potentially important aspect of environmental negotiations, namely its distributive implications. The present paper tries to fill this gap assuming that countries\regions have Fehr and Schmidt (1999) (F&S) utility functions, specifically tailored for including inequality aversion. Thereby, we propose a new method for the empirical estimation of the inequality aversion parameters by establishing a link between the well known concept of elasticity of marginal utility of consumption and the F&S utility functions, accounting for heterogeneity of countries/regions. By adopting the RICE model, we compare its standard results with the ones obtained introducing F&S utility functions, showing that, under optimal cooperation, the level of temperature rise is significantly lower in the last scenario. In particular, in the last year of the simulation, the optimal temperature rise is 2.1◦ C. Furthermore, it is shown that stable coalitions are easier to be achieved when F&S preferences are assumed, even if the advantageous inequality aversion parameter (altruism) is assumed to have a very low value. However, self–sustaining coalitions are far from reaching the environmental target of limiting the mean temperature rise below 2◦ C despite the adoption of F&S utility functions.


Introduction
including the elasticity of marginal utility (EMU) of consumption into the countries\regions 48 utility function. Actors with larger levels of per-capita consumption will enjoy lower levels 49 of utility increase for additional units of consumption. However, we argue that this standard 50 formulation of a concave utility function in per-capita consumption may not be adequate 51 to fully capture the disutility caused by inequality. 52 In particular, in the present paper we propose a more systematic inclusion of equity 53 concerns based on insights from behavioural economics in the form of Fehr and Schmidt 54 (1999) (F&S) utility functions. In fact, these capture the phenomenon that people compare 55 themselves to others and possibly derive dis-utility if their payoff is below or above other 56 players' payoffs. This utility function is in line with numerous observations made in exper-57 imental economics and it has proven to be successful in explaining observed behaviours in 58 bargaining and cooperation games (see Fehr and Schmidt (2006) for a review).

59
F&S utility functions have already been applied to the problem of voluntary agreements 60 in international climate policy by Lange and Vogt (2003), Vogt (2016) and Rogna and Vogt 61 (2020). However, the focus in these papers has primarily been to analyse the effect of 62 inequality aversion on the prospects of voluntary cooperation via the means of coalitions. 63 Furthermore, they all share the use of very simplified and atemporal models, typical of 64 a game-theoretic analysis, that are not suitable to derive predictions about environmental associated temperature rise, has remained far above the targets of 1.5 • \2 • C settled in the Paris Agreement. Note that this is not a peculiarity of DICE\RICE, since several other 112 models provide an estimate of the SCC that is inadequate to meet the 1.5 • \2 • C targets, 113 as shown in Tol (2019) and in Ackerman and Munitz (2016). This has generated a wave of 114 critiques directed towards IAMs. As mentioned in the introduction, the underestimation of 115 catastrophic and immaterial damages (Weitzman, 2012;Howard and Sterner, 2017), the lack 116 of account for uncertainty (Roughgarden and Schneider, 1999;Diaz and Moore, 2017) and 117 a high inter-temporal discounting (Dietz et al., 2018) are the main targets of the mentioned 118 critiques. 119 Several attempts have been made to overcome these perceived shortcomings. De Bruin  With its disaggregation into 12 countries\regions, the RICE model has also been used 131 to investigate the stability of international environmental agreements (IEAs), depicted as 132 coalitions. The results, in line with other numerical models such as WITCH (Bosetti et al.,133 2006) and STACO (Dellink, 2011), have confirmed the grim predictions of early game-134 theoretic analyses -e.g. Carraro and Siniscalco (1993) and Barrett (1994) -, namely that 135 only few and relatively small coalitions are stable (Yang et al., 2008).

136
On the game-theoretic side of climate change analysis, there is a number of papers that  The present paper aims at including non-standard preferences into a dynamic and more 147 complex model framework such as the RICE model, thus filling a current gap of the liter-148 ature. The choice is for F&S preferences whose consideration of aversion for both advan-149 tageous (altruism) and disadvantageous (envy) inequality has proven to be able to capture 150 several deviations from standard economic theory observed in laboratory experiments (Fehr 151 and Schmidt, 2006). In particular, we are interested in observing which is the effect of this 152 alternative specification of the utility function both on the optimal level of emissions abate-153 ment, and, therefore, on the temperature rise, and on the stability of climate coalitions. Our starting model, also used as benchmark, is RICE v2013. A synthetic description of all 156 its variables (endogenous and exogenous) and parameters can be found in the Appendix, thermore, leaving a single control variable, abatement, simplifies and renders more explicit the interpretation of results.

167
The second modification refers to the discounted utility function of countries -equa-168 tion (A2) in the Appendix -whose numerator, differently from the original version, is not 169 multiplied by the population size (L i,t ). This term will then act as a weight when summing 170 utilities in coalitions, with more populous countries\regions gaining more importance. It 171 may actually be reasonable and realistic to have such term since, in a bargaining process, 172 larger countries could effectively hold more bargaining power. However, the F&S prefer-173 ences only consider per-capita consumption when operating the inter-countries comparison.

174
In order to keep as close as possible the two types of utility functions that will be com-175 pared, it seems then opportune to drop the population weight. Furthermore, this drop can 176 be theoretically justified by the fact that countries\regions, being sovereign entities, act as 177 individuals and the "power" granted by a larger population size is hardly quantifiable, if 178 justifiable at all.

179
Except for the two modifications just explained, the set of equations in section A2 in the Appendix faithfully reproduces the original RICE model v2013. This will be used as our benchmark scenario, without adding any exogenous environmental target or any price for CO 2 . As mentioned in the introduction, our main assumption is that, in order to properly capture the relational component of utility arising from comparing the own level of economic attainment (per-capita consumption) with the one of the others, the marginal utility of consumption -η i in equation (A2) -is not sufficient. The F&S utility function, instead, is better suited for this purpose. Following is the mathematical definition of the F&S utility function: where i is a generic player of set N , whose cardinality is represented by n, π is the payoff 180 of a player, I + and I − are the sets of players having, respectively, a payoff higher and 181 lower than player i and, finally, α i and β i are the parameters representing the aversion for 182 disadvantageous, the former, and for advantageous, the latter, inequality. The expression 183 α i n−1 j∈I + (π j − π i ), where the component inside the round brackets is always positive since 184 π j > π i by definition, represents the disutility suffered by player i for having a payoff lower 185 than all players j (envy). Similarly, the expression following β i , necessarily positive by 186 definition as well, represents the disutility for advantageous inequality (altruism).

187
Willing to adopt the F&S utility function in the RICE v2013 model, equation (A2) in the Appendix must be substituted by the following two equations: where π i,t is simply defined as the discounted value of per-capita consumption. While all 188 the parameters of the model can be retrieved from the documentation of RICE, the addition 189 of the new utility function brings the burden of estimating α and β. The next sub-section 190 is dedicated to describe the procedure adopted for retrieving them.

191
The estimation of α and β

192
Since the values of α and β represent the intensity with which the disutility from disadvan- be culturally influenced, thus determining important inter-country differences.

219
In order to partly remedy to these shortcomings, we attempt to estimate the values of both α and β is precluded since there are too many unknowns for the system of equations 223 that will be used, therefore we still need to assume a value for the βs. Furthermore, given 224 the need of assuming such values, we have no indication on how to vary them among the 225 countries\regions, therefore we prefer to retain the assumption of homogeneity. The choice 226 of estimating α and not β is due to the fact that, in Rogna and Vogt (2020), the former has 227 a much more important role in determining the stability of coalitions. Furthermore, pure 228 altruism at country level seems less likely to exist than dissatisfaction for disadvantageous 229 inequality, therefore its magnitude may be very small, leading to a negligible role.

230
It is important to remind here that F&S utility functions address the underlying causes of inequality aversion, namely interpersonal income comparisons, which are not captured 232 by the concept of EMU. This has an important consequence: F&S utility functions capture 233 the psychological externalities additionally to the classical, environmental external effects.

234
With EMU, instead, only the diminishing marginal utility of consumption for higher levels 235 of affluence is captured, leaving aside the psychological effect of inequality. This may have 236 a significant impact on the optimal level of abatement and, consequently, on the level of 237 temperature rise.

238
The reasoning behind our estimation procedure is as follow. Consider the definition of elasticity of marginal utility of consumption: where c is consumption. In our definition of F&S utility we have defined π i,t as the discounted value of per-capita consumption. Let us drop, for mere convenience, the temporal dimension and equate π i to c i . Given the F&S utility function as in equation (1), the marginal utility of consumption (for the sake of brevity we avoid to repeat per-capita consumption) is simply given by: where |I + | and |I − | indicate the cardinality of the sets of players with a level of consumption higher and lower than player i, respectively. Now, if we consider a discrete increase in the consumption of player i such that she switches of one position in the consumption ranke.g. |I + | will be decreased by one unit and, consequently, |I − | will be increased by the same amount -, we can define dM U i dc i as: where, for sake of brevity, c 2 i − c 1 i = ∆ i . Note that we have α i+1 and β i+1 since, at level of consumption c 2 i , player i shifted up of one position in the consumption ranking. By dividing c i for the RHS of equation (3) and multiplying the result to the RHS of equation (4), we get EMU i :  (1999) given that we assume countries to be 245 less altruistic than individuals, is adopted. However, a further assumption is required. Due 246 to the fact that I + = ∅ for the most affluent country (US in the RICE database), we are 247 basically left in shortage of one equation since α i |I + | = 0 for this country. Therefore, also 248 this value of α must be assumed and we have set α U S = 0.11. In Table 1 Actually, we report the results till the year 2100, but we have run the simulation till 2110 to reduce the last periods drop in abatement consequent to a "no future" scenario.
3 The model and the data used for the simulation are available on a GitHub repository at this link.
, the condition reads as: If this inequality is satisfied for each member of coalition C, than C is internally stable.

343
Our numerical simulation has fully confirmed this result, that, besides providing a method 344 to check for coalition stability, is also useful to understand the reasons inducing to internal  Besides leading to a temperature rise close to the 2 • C threshold in the cooperative case, 360 that is, however, not stable, the F&S scenario with heterogeneous αs allows for more sta-361 ble coalitions and for even more coalitions that can be fully stabilised through transfers.

362
However, the simple number of stable coalitions may be considered as scarcely informative, 363 since it does not tell much about the outcomes. In particular, it is interesting to see, in 364 the various scenarios, which is the best and realistically achievable target. By adopting 365 an environmental perspective, we consider as best the coalition that allows for the lowest 366 temperature rise in the last period, and as realistically achievable a coalition that is stable 367 or potentially stable. Clearly, in the baseline scenario, since there are no stable coalitions, 368 we have the same outcome described earlier, with a temperature rise of 3.24 • C in 2100.

369
Even in absence of cooperation, this is a lower temperature rise than the one obtainable  In this last section, the sensitivity of the results to variations in the α and β parameters 396 is tested. We focus on the two extreme cases, the grand coalition and the total absence of 397 cooperation, looking at the end of period temperature rise for different values of the homo-398 geneous βs and for an alternative assumed α value for US. Table 3 reports the estimated 399 α values under the two different assumptions. By comparing Table 3 with Table 1, it is 400 possible to observe that a different value of α for US, now equal to 0.833 compared to 0.11 401 adopted in the previous simulations, does not cause a dramatic change in the estimated α 402 values of the remaining players, many of which are actually identical.

403
It comes with no surprise, therefore, that also the environmental outcomes of the new 404 simulation do not differ substantially from the ones obtained before. In the cooperative 405 case, the temperature rise in 2100 is equal to 2.21 • C, meaning 0.11 • C higher than in the 406 simulation with α of US equal to 0.11. With regard to the non-cooperative case, the final 407 temperature rise is of 3.30 • C, thus identical to the previous simulation with heterogeneous 408 αs.

409
It is interesting to notice that, even under a different value of β for all countries (0.2 410 compared to 0.1 in previous simulations), that causes a significant variation in the esti-411 mated α values, as can be seen from Table 3, the outcomes are still almost unchanged.

412
The temperature rise in the cooperative case, in fact, is still equal to 2.21 • C, while it is 413 equal to 3.29 • C in the non-cooperative case, a mere difference of 0.1 • with previous results.

414
The outcomes of the simulation, therefore, are quite robust, at least to relatively small per-415 turbations in the parameters governing the aversion to advantageous and disadvantageous 416 inequality. are scarcely affected (see Table A31). We can notice an increasing level of temperature rise 425 for increasing values of US α both in the cooperative and in the non-cooperative case, but 426 this rise is so subtle -at the fifth decimal place -that can be considered as irrelevant.

428
The economic-environmental models that have been used to investigate the opportunity-429 costs of limiting the temperature rise caused by greenhouse gases have been often criticized 430 for suggesting low levels of CO 2 emissions reduction. A very well known example is one of the first adopted models to undertake this type of analysis, the DICE\RICE model, whose 432 predicted optimal path leads to an increase of roughly 3 • C over the pre-industrial mean