Climate change control: the Lindahl solution

Article 2 of the 2015 United Nations Climate Change Conference Paris Agreement aims at limiting global warming to well below 2 °C and “Making finance flows consistent with a pathway towards low greenhouse gas emissions and climate-resilient development.”¹ Contrary to the Kyoto protocol, there is no detailed country- and time-specific path to reach these goals, but instead a bottom-up approach where each country can set its own “nationally determined contributions” to cut greenhouse gas (GHG) emissions, without an enforcement mechanism. Apparently, the specifics of burden sharing is relegated to the 5-yearly periodical global stocktake specified in Article 14, starting in 2023, although Article 4 states that developed countries should make absolute reductions, against for the near future just mitigation efforts for developing countries and of mobilizing USD $100 billion per year to fund climate policies financed by the developed countries.

Long before Paris 2015, there has been an ongoing and flourishing debate among scientists about the time path to contain global warming. The timing issue is largely concerned with the choice of appropriate discount rates (Weitzman 2001; Nordhaus 2007; Heath 2013) and how to assess the risk of unlikely but potentially disastrous outcomes, e.g., reversal of the “thermohaline circulation,” also known as the “Great Ocean Conveyor Belt” of the Gulf Stream and the exacerbated release of methane due to global warming from the Arctic permafrost. Here, we abstract from the time path and concentrate on how the abatement burdens should be allocated across regions in the world at a given point in time. Given the urgency to reduce GHG emissions worldwide, the most difficult issue is how the burdens are distributed across regions. Since climate change is a global public good, it requires a global burden sharing rule. The two dominant guiding principles for fair burden sharing are the polluter pays principle and the ability to pay principle. For instance, the Stern Review (Stern 2007, p. 23) states that based on income, historic responsibility, and per capita emissions, rich countries should take the primary responsibility to combat climate change. However, the literature on burden sharing emphasizes the free-riding problem as a consequence of climate change being a public good (Gupta 1997). Thus, one important strand in the literature has analyzed the challenge of relating damage caused by climate change to economic activities, which are often occurring in distinct places (see, for example, Edwards and Miller 2001; Voigt 2008 and Dellink et al. 2009). This paper contributes to this literature by avoiding the problem of having to establish the causation from polluter to damage experienced, by turning instead to countries own interest. In this paper, we derive the burden sharing rule following the Lindahl equilibrium (hereafter LE), where burden shares are not so much determined by ability to pay or (historical) emissions, but by countries’ interest, e.g., due to expected damages, to combat global warming.

To illustrate the LE by means of a simple example, suppose two persons share a household in which the cleanliness of the house is considered a public good. The problem of burden sharing is how many hours every member of the household has to spend on cleaning. A simple 50–50 split will not do, because one person may prefer to have the kitchen or closet much cleaner than the other, so even at first glance, fair 50–50 split will not solve the question how many hours in total will have to be spent on cleaning, that is, the provision level of the public good. The LE will identify a unique level of the public good with shares assigned to each household member in such a way, that given the assigned shares, each member will choose the same (Lindahl) level of public good provision with the shares summing up to unity. However, since shares are proportional to marginal willingness to pay (WTP), household members have an incentive not to reveal their true preferences in order to easy- or free-ride on the efforts of the other.

Although one may question how plausible it is that people durably living in a household are able and willing to hide true preferences to get an advantage in burden sharing at the expense of the other member, with many agents sharing a public good strategic misrepresentation of preferences can indeed be a serious problem. With respect to an agreement involving many states, the identified LE from the scientist’ drawing table is therefore only useful as long as the analysis is based on easily observable variables such as GDP, population size, GDP per capita, energy consumption, GHG emissions, and to be expected damages from climate change that are not easy to manipulate.

In the burden sharing debate of climate change, surprisingly little attention is paid to the LE.² In the LE, each country is assigned an abatement share in such a way that given their assigned share, they all want the same level of the global public good—in this case global carbon emission abatement (for the derivation of Lindahl equilibria in public goods models, see Sandler and Murdoch 1990; Mas-Colell 1989; Shitovits and Spiegel 1998, 2003). In such a LE, each country contributes to the global provision level according to its WTP. The saliency of WTP is that any International Environmental Agreement in which some countries have to contribute less than their WTP is a pity, because they are prepared to abate more. At the same time, a country will be reluctant to contribute in excess of its WTP, for instance, if it is prescribed to do so by invoking the polluter pays’ principle or ability to pay principle. Therefore, any deviation from the LE would mean that either some countries contribute less than their actual WTP, which is undesirable if we are to realize the aim of mitigating GHG emissions, or that some countries are supposed to contribute more than they are willing to, making the agreement unstable. Therefore, it is important to assess the burden sharing rule according to the LE and to check how it fares compared to other burden sharing rules.

Buchholz and Peters (2007, 2008) have identified the main fairness properties of the LE. They show not only that the LE is efficient (i.e., satisfying the Samuelson condition for the optimal supply of the public good) but also that the benefit principle (described as “everyone pays what he gets”) and the axiom of proportional contributions (meaning that cost shares are proportional to marginal WTP) are satisfied. Despite these attractive properties, the neglect of the LE should not come as a surprise, since in the economic literature on burden sharing to provide a public good, the LE is said to be merely of theoretical interest, mainly for two reasons: first, the difficulty to assess objectively for each country its WTP and second, the incentive to strategically misrepresent preferences. The best illustration of the practical insignificance of the LE in the debate on climate change is that as far as we know, there is no study with an empirical simulation of the abatement burdens across countries or regions in a LE, a lacuna we hope to fill in this paper.

The structure of this paper is as follows. Section 2 presents the sub-optimal non-cooperating Nash model. Section 3 presents the model with a social planner but without the instrument of redistribution, which, together with the Nash model, are used as benchmarks for the LE. Section 4 identifies the LE in the global burden sharing of abatement. The empirical part is presented in Section 5. In the first simulation, the world is divided into two blocks of rich (Annex I) and poor (Annex II) countries; in the second, five regions are distinguished. The different burden sharing rules are evaluated by two criteria, the extent to which global abatement is optimal and the required degree of redistribution by means of transfers from rich to poor countries. The final section summaries and concludes.

In modeling climate change and abatement, many choices have to be made. Does one take a static or a dynamic perspective, is the externality arising in consumption or production or both, is the approach rooted in welfare economics or game theory, what is the appropriate discount rate, and so on. In general, many of the choices made here are motivated by keeping the model as simple as possible in order to derive burden sharing rules of abatement under different regimes. In the first regime, serving as a benchmark for the models presented in subsequent sections, countries are assumed to follow their self-interest in a non-cooperative way. Following the withdrawal of the USA on 1 June 2017 from the Paris agreement, ratified on 5 October 2016 and due to the absence of a burden sharing rule and enforcement mechanism, it is not exaggerated to claim that the Nash model, where each country pursues its own interest taking into account the behavior of others, might still be an appropriate workhorse carrying a sense of realism in case the bottom-up Paris approach fails. Cramton and Stoft (2010) go so far to say that after Kyoto, “In fact there is no clear evidence that we have done even as well as the public-goods Nash equilibrium.” First, we model non-cooperative behavior without and with a permit market.

Now, suppose countries agree to install a social planner (labeled S) to redress the sub-optimalities of the Nash outcome. If S is given not only the power to set the burden sharing rule for abatement but also the power to redistribute income, the global welfare maximizing outcome will be equality of marginal utilities of income across countries and uniform marginal cost of abatement to ensure production efficiency (see Appendix 1). Although equity and efficiency are achieved simultaneously, it is not realistic to assume that in order to solve the global warming problem, however serious it may be, sovereign rich countries are prepared to equalize their per capita incomes to that of the rest of the world. Therefore, a more realistic version of S is that it lacks the instrument of income redistribution but is still given the restricted mandate to devise an optimal burden sharing abatement rule. We will see that the optimal rule requires the marginal welfare cost of abatement to be equal across countries. We distinguish S without and with the power to install a global permit market.

From a moral point of view, the acceptability of the Lindahl solution is hampered because it is in conflict with both the polluter pays principle and ability to pay principle. Gardiner (2004, p. 590) concludes that: “… there is a great deal of convergence on the issue of who has primary responsibility to act on climate change. The most defensible accounts of fairness and climate change suggest that the rich countries should bear the brunt, and perhaps even the entirety, of the costs.” In the same vein, the Executive Summary of the Stern Review (Stern 2007, p. 23) states that “Securing broad-based and sustained co-operation requires an equitable distribution of effort across both developed and developing countries. There is no single formula that captures all dimensions of equity, but calculations based on income, historic responsibility and per capita emissions all point to rich countries taking responsibility for emissions reductions of 60–80% from 1990 levels by 2050.” Apparently, both Gardiner and Stern favor a burden sharing rule in which the rich countries bear the lion share of the costs due to their higher ability to pay and to the polluter pays principle.

However, assessing burdens to combat climate change is not merely a morality play. Countries are sovereign and a burden sharing rule based on moral principles—such as the polluter pays principle or the ability to pay principle—is only as strong as the commitment of countries to these principles.¹⁰ The ability to pay or paying as polluter may not be in line with a countries’ willingness to pay.¹¹ Our proposed burden sharing rule is based on countries’ willingness to contribute to combat climate change, which naturally leads to the Lindahl solution to the optimal public good provision level. The Lindahl solution has two major disadvantages. Firstly, the willingness to pay is not only influenced by the expected damage but also by the adaptation costs, which puts many poor countries in a precarious position if they lack the resources and know-how for adaptation. Secondly, contributions according to willingness to pay may be squarely at odds with contributions based either on the ability to pay and the polluter pays principle. For the sake of argument, suppose that the USA is protected from any consequences of climate change and that only the rest of the world would suffer damages. According to ability to pay and the polluter pays principle, the USA would have to contribute heavily, but its willingness to pay might be low. Analogous, the burden assigned to a poor country at sea level according to ability to pay and polluter pays will be low, but relatively high according to willingness to pay. In theory, countries that would benefit from global warming (e.g., Russia, Canada, and Greenland) may have a negative willingness to pay, which implies that they have to be compensated for their participation in an international agreement to abate greenhouse gases. This suggests that concerns of redistribution between rich and poor should be excluded in considerations of how to assess fair burden sharing to combat climate change. What we propose is therefore that climate change policy must be evaluated on its own merits, and the same goes for policies to address global income inequality (see, e.g., Milanovic 2016).

In this section, we compare the outcomes of different regimes of burden sharing rules, especially with respect to the extent that optimal global abatement levels are achieved and to the amount of transfer payments relative to the total cost of abatement. For each regime, we measure the level of abatement relative to the optimal abatement level under lump sum (LS; see Appendix 1). The operationalization of the equilibrium conditions for each regime are explained in Appendix 2. We measure redistribution by the share of the transfer payments made by countries with a higher target abatement level than their actual abatement in global abatement costs. It is measured as

TP is an indicator of the share of global abatement costs financed by other countries’ payments on the permit market. For regime LS, in which per capita incomes across countries are equalized and where it does not matter who pays how much, we set TP equal to unity to express that all costs of abatement are shared. If there is no permit market and every country finances its own abatement, the indicator TP is zero by definition.

We also calculate which part of the total cost of abatement is shouldered by the rich countries¹⁵ (e.g., Annex I, or Europe, Oceania, and North America in case of the five regions; see below), measured as

Compared to TP, the measure TC also includes actual abatement costs. If there is no permit market, then it simply measures the abatement cost shares.

We have chosen the parameter α for the relative importance of abatement relative to consumption and the cost parameter c (see Appendix 2) so that the simulation results simultaneously yield plausible marginal abatement cost (in the range of $20 to $80 per tonne CO₂), total abatement cost as a share of GDP, and total abatement efforts (e.g., the Stern Review (Stern 2006) recommends a significant reduction of 60–80% by the rich countries in 2050 relative to 1990).¹⁶ All data are for 2014 and obtained from CAIT Climate Data Explorer (Washington DC, World Resources Institute 2016) and from EDGAR (2009, European Commission Joint Research Centre, Netherlands Environmental Assessment Agency). Table 1 gives an overview of the scores on salient variables if the world is divided into only two blocks, Annex I and Annex II. The Annex I countries comprise the regions Europe and Oceania (EU) and North America (NA). The non-Annex I countries comprise the regions sub-Saharan Africa and Middle East and North Africa (AF), South America and Central America and Caribbean (SA), and Asia (AS). In Table 2, the results are presented if the world is divided into five regions.

The top panel in Table 1 contains the descriptive statistics: population (in millions), GDP (in billions USD), income per capita, emissions (in Mt.), emission per capita (in tonnes), emission intensity, and the marginal cost adjustment parameter r (see Appendix 2, Eq. 26). Income per capita in Annex I countries is more than seven times as high as in Annex II countries and emissions per capita is more than three times as high. In the second panel, (target) levels of abatement and reduction rates are given, where we take the actual emission levels in 2014 as the business-as-usual outcome (which implies that the Nash outcome is the one where Annex I and I operate as blocks). Firstly, total abatement with a permit market under the same regime is always higher or equal (total abatement under S and SP are set equal by assumption) than without. Secondly, in the shift from N to NP, the Annex II target abatement level under Nash with a permit market is even below its actual abatement without a permit market. This result can be explained by polarization, also described by Cramton and Stoft (2010, p. 6), where the rich will choose an even higher target level because abatement can be bought more cheaply under a permit market, while the poor region will choose an even lower target level of abatement to benefit from permit trading. Thirdly, total abatement under the social planner, without (S) or with (SP) a permit market, is higher than under Lindahl (L). This is because assigning abatement burdens has a dual role for the social planner SP, not only to mitigate global warming but also to redistribute income or welfare. In case of S, Annex I is assigned a very high abatement burden (a reduction rate of 84%, against only 47% for Annex II), because the welfare cost of abatement for the rich countries are relatively small, while the benefits of abatement are global.

The third panel gives information about per capita incomes (y), utility (u), world welfare (W), the permit price (q), average abatement cost (AC), the share of total abatement cost paid for by transfers (TP), and the share in total cost of buyers on the permit market (TC). Not surprisingly, global per capita welfare is at maximum in the lump sum case, but it would not be acceptable for Annex I. Departing from regime N, Annex I would even not be in favor to move to regime NP (due to the polarization effect), nor to S or SP. The only transitions that increase utility for Annex I are the Lindahl regimes L and LP. For Annex II, all other regimes than N are better in utility terms, where S and SP are preferred to L and LP. Taken together, departing from N, only L and LP are Pareto improvements and LP Pareto dominates L, so LP would be a viable outcome.¹⁷ The equilibrium permit price under LP is 0.077 billion per Mt. CO₂, which corresponds to $77 per tonne, while average abatement cost per tonne is $34 (due to increasing marginal costs of abatement, average cost is below marginal cost).

The last panel gives total abatement relative to (optimal) abatement (A), the share of total abatement cost financed by permits (TP), and the share of the total cost of abatement taken care of by the rich countries (TC). The first two of these measures are illustrated in Fig. 1a, b. Apart from LS, there are four regimes that deliver abatement equal or close to the optimal level under LS. Production efficiency requires the regimes with permit markets. Among the permit market choice set {NP, SP, LP}, LP combines that abatement is at the optimal level and transfer payments as a share of total costs are at minimum. Taking stock, LP (L) are the only regimes that are Pareto-superior to N and combining (near) optimal abatement levels with modest (zero) redistribution payments.

In Table 2, the world is divided in five blocks. CAIT standardly provides a division into the eight geographical regions, Asia (AS), Europe (EUR), Middle East and North Africa (ME), sub-Saharan Africa (AFR), North America (NA), Central America and Caribbean (CAM), South America (SAM), and Oceania (OC). In Table 2, ME and AFR are merged into AF, CAM, and SAM into SA and OC and EUR into EU. In this division, two are rich (North America and Europe with Oceania) and two are poor (Asia and Africa), with Central and South America in between.

Because there are now more players, total abatement levels under Nash are almost halved compared to when Annex I and II operate as blocks. Departing from N, again only L and LP are Pareto improvements for all regions and LP Pareto dominates L. A striking outcome in Table 2 is that for both Africa and South America, it is optimal to choose negative target abatement levels. The extreme polarization leads here to the situation that under NP total abatement is even lower than under N, despite the efficiency gains of a permit market. Note that under regime NP, each player is free to choose its optimal target level. Net revenues from the permit market equals q(A _i  − T _i ), so although South America abates 240 Mt., by choosing a target level of − 3333 Mt. and selling permits for the equilibrium price of $15.6 per tonne, it receives $55.7 billion on the permit market, while Africa receives $60.6 billion.¹⁸ Asia (due to its high population), North American, and Europe and Oceania (due to their high per capita incomes) together pay in total 116.3 billion (their combined target abatement levels of 13,022 Mt. minus their combined actual abatement of 5558 Mt., times $15.6 per tonne), whereas total abatement cost is only 49 billion (total abatement under NP equal to 6451 Mt. times average cost of $7.6 per tonne). The main reason for the outlier position of regime NP in Fig. 2a, b is that transfer payments under NP outweigh total abatement cost, which is not so much due that the average cost of abatement is below the permit price or marginal cost, but because of polarization.

Not surprisingly, total reduction levels (R) for the other regimes are almost equal compared to under Annex I and II. Apart from the location of N, shifting downwards and NP, shifting downwards and to the right, Fig. 2a, b therefore gives the same configuration of regimes as in Fig. 1a, b, where regime LP combines optimal abatement with modest transfer payments as a percentage of total cost.¹⁹

The Paris agreement of 2015 entails that both developed and developing countries limit their emissions, with 5-yearly reviews to ensure that national contributions are in line with the overall goal to reach at maximum 2 °C global warming, with an aspiration of 1.5 °C. In this paper, the abatement burden sharing rules emerging under different regimes, with and without permit markets, were derived. The sub-optimal level under Nash without permit market is due to both production inefficiencies in abatement and not taking global externalities of national abatement into account. The first sub-optimality can be removed by installing a permit market, but the second requires cooperation or coordination between countries. A social planner without the power to redistribute and without a permit market will impose the rule that the product of the marginal cost of abatement and the marginal utility of income be equalized across countries. This implies high abatement burdens and (marginal) costs for rich and low burdens and costs for poor countries, with the overall abatement level close to optimal. The same social planner but equipped with a permit market can organize transfer payments from rich to poor by assigning high target levels to rich and low target levels to poor regions. The burden sharing rule then has a secondary role of redistribution and without any constraint the social planner’s outcome will be the same as under an omnipotent social planner with lump sum redistribution.

In the simulations, we showed that the transition to a permit market under Nash can lead to polarization, eventually leading to lower overall abatement as shown for the world divided into five regions. Although poor countries prefer the social planner regime with permits the most, only the Lindahl regimes Pareto dominate the Nash regimes, with or without a permit market. Moreover, of all permit regimes, the Lindahl permit market entails the lowest degree of redistribution in the form of net transfer payments from rich to poor. Our preferred burden sharing rule can be summarized as that every country or region shares in the burden to combat climate change in proportion to its benefits, which are determined by the expected damages and adaptation costs. The burden sharing rule implied by the Lindhahl solution therefore provides a criterion to assess the national contributions to curb climate change. The Lindahl solution simultaneously achieves an optimal global abatement level and that each country is expected to contribute according to its willingness to pay, but the price to be paid is to disregard competing principles such as ability to pay and the polluter has to pay. In our analysis, we only differentiated countries by their income per capita and population size. Avenues for further research are to relate the Lindahl solution and the corresponding abatement burdens to country- or region-specific expected damages and associated adaptation costs from climate change as proxies for its willingness to pay (e.g., using an integrated assessment model such as RICE) and a more systematic comparison how the Lindahl solution fares compared to other fairness principles governing burden sharing rules for global public goods.