The following assumptions define the relevant framework conditions of our model. These first assumptions characterize countries as members of a union to mitigate climate change, the individuals living in these countries and the relevant commodities. The basics of our model correspond to the model of Bergstrom et al. (1986).

Assumption 3.1.

a) There is a set N = {1, …, n} of countries. N constitutes a union of countries pursuing the environmental goal of mitigating climate change. There are k_i individuals in country i, i ∈ N.

b) There is one private commodity x and one public commodity y. In the context considered here, x is the gross domestic product (GDP), available for private consumption. The public commodity y is represented by the benefits of contributions to renewable energies (measured through renewable energy consumption).

c) Individuals in country i, i ∈ N, are characterized by the identical initial endowment w_i of the private commodity (thus, GDP per capita), and the identical utility function depending on consumption of the private commodity x and the public commodity y. For each i ∈ N, utility is given by the homothetic function u_i (x_i , y) := x_i y^α_i with the “awareness” parameter α_i > 0.

“Renewable energy consumption” is used as an indicator regarding efforts to mitigate climate change by reducing greenhouse gas emissions. This public commodity is provided through the employment of renewable energy sources in the various countries, in our case parties to the Kyoto Protocol.

The parameter α_i is closely related to the marginal rate of substitution between the private and the public commodity. In fact,

MRS_i (x, y) = u_iy (x, y) / u_ix (x, y) = α_i (x / y) (1)

for an arbitrary consumption bundle (x, y) ∈ IR²₊₊. Therefore, a higher value of α_i indicates ceteris paribus a higher “willingness to pay” for an additional unit of the public commodity. In this sense, the values α_i, i ∈ N, can be considered as indicators of “awareness” for global warming (cf. also Diederich and Goeschl, 2013), which allow a ranking of the countries.

The next assumption refers to the production possibilities of the public good, i.e., to the costs of producing 1 kWh of electrical energy by means of renewable sources. There are, of course, cost differences for the various renewable energy sources (cf. Fraunhofer ISE, 2013), and concrete costs depend on the combination of these sources, which varies a lot across countries, also due to geographic and climatic conditions. Moreover, advanced technologies to generate electricity from renewable sources are more likely to be used in industrialized countries, and costs will also differ due to differences in wage rates and prices for suitable lots. In order to estimate the costs of generating electricity from renewable sources, we use data on “levelized cost of electricity” (LCOE).2 We make the following assumption, which will be complemented with concrete numbers for LCOE in Section 4.

Assumption 3.2. In country i, i ∈ N, β_i units of the private good can be turned into one unit of the public good. Thus, each country has access to a technology with constant returns to scale to produce the public commodity. β_i should be understood as the average LCOE according to the mix of renewable sources applied in country i.

Then utility of, for example, individual 1 of country i, i ∈ N, can be rewritten using the contributions t_j^m, j = 1, …, k_m, m ∈ N, of all individuals towards the provision of the public good:

v_i (t₁¹,…, t₁^k1; …; t_n¹,…, t_n^k_n) :=

= (w_i – β_i t_i¹) ∙ (t₁¹ + … + t₁^k1 + … + t_n¹ + … + t_n^k_n)^α_i =

= x_i¹y^α_i = u_i (x_i¹, y) (2)

We make the following assumption with respect to the utility-maximizing behavior of the individuals in each country i, i ∈ N:

Assumption 3.3. Individual agents maximize utility given the actions of all other agents in all countries.

Although decisions on the application of renewable energy sources are often initiated and stimulated by governments, individual households or companies play an important role in this context.3 In addition, governments cannot consistently and over a longer period of time neglect the preferences of the voters. This supports the behavioral assumptions of this model. However, it is also possible to assume that individual countries are the decision-makers. This leads to slightly different, but not structurally different results.

We then obtain the following first order condition for individual 1 in country i ∈ N:

β_i (t₁¹ + … + t₁^k1 + … + t_n¹ + … + t_n^k_n) = α_i (w_i – β_i t_i¹) (3)

As the left-hand sides of these first order conditions for the individuals of country i are identical, the right-hand sides must be identical, too, resulting in identical equilibrium contributions of all agents of this country. Thus t_i¹ = … = t_i^k_i =: t_i in equilibrium for each i ∈ N. Consequently, the first order conditions for i ∈ N can be rewritten as follows:

k₁t₁ + … + (k_i + α_i) t_i + … + k_n t_n = α_i (w_i / β_i) = α_i w^{^}_i (4)

with “real” income w^{^}_i := w_i / β_i measured in kWh of electricity from renewable sources.

As already indicated, the Nash mechanism is certainly among the most prominent approaches towards describing the interactions of the countries or, rather, the individuals, regarding the provision of this particular public good. Other forms of interactions, leading to, for example, egalitarian-equivalent allocations or core allocations (cf. Moulin 1987, Weber and Wiesmeth 1991a), require a more intense cooperation among the partner countries, which, in general, can only be guaranteed by a supranational institution endowed with sufficient administrative power. There is no such institution for the cases considered here.4

In the next step, we look for the solution, the Nash equilibrium, resulting from these interactions via the Nash mechanism. We thereby restrict the analysis to the consideration of interior solutions, which are relevant in most practical situations. For all cases we use real income w^{^}_i := w_i /β_i, i ∈ N. As already mentioned, under Assumption 3.3 the first-order conditions for an interior solution are given by: k₁t₁ + … + (k_i + α_i) t_i + … + k_n t_n = α_i w^{^}_i for each i ∈ N.

We then obtain the following proposition for the values t = (t₁, …, t_n) ∈ IR²₊₊ for the Nash-equilibrium. The proof of Proposition 3.1 is provided in the Appendix.

Proposition 3.1. First of all, the solution t = (t₁, …, t_n) ∈ IR²₊₊ is symmetric in the sense that t_i can be obtained from t_j by replacing in t_j each occurrence of the index j with the index i and vice versa. Moreover, t₁ is given by:

t₁ = [k₂ (α₁α^{^}₂ … α_n) w^{^}₁ – k₂ (α^{^}₁α₂ … α_n) w^{^}₂ + … + k_n (α₁α₂ … α^{^}_n) w^{^}₁ –

– k_n (α^{^}₁α₂ … α_n) w^{^}_n + (α₁ … α_n) w^{^}_n] /

/ [k₁ (α^{^}₁α₂ … α_n) + … + k_n (α₁ … α^{^}_n) + (α₁α₂ … α_n)] (5)

with α^{^}_i meaning that this factor has to be replaced by 1. By using α^–_i := (α₁ … α^{^}_i … α_n) and α^– := (α₁α₂ … α_n) , we can simplify this expression in the following way:

(6)

Again, we assume that an interior solution with t₁> 0, i ∈ N, exists for the given constellation of the parameters.

Returning to individual equilibrium contributions t_iⁿ and total equilibrium contributions T = k₁t₁ + … + k_n t_n , we arrive at the following results, which follow immediately from Proposition 3.1 (remember that w_i is given by GDP_i per capita, i ∈ N) :

Result 3.1. A higher awareness for the public good results cet. par. in a higher individual contribution towards the provision of the public good. Similarly, a higher GDP per capita results ceteris paribus in a higher individual contribution towards the provision of the public good. Moreover, total contributions T increase with increasing w_i of the participating countries and also with higher “awareness”, i.e., with increasing values of α_i , i ∈ N.

The following result is a consequence of the neutrality theorem of Bergstrom et al. (1986) applied to our context. The proof is provided in the Appendix.

Result 3.2. Assume that β_i > β_j for i, j ∈ N and consider a monetary transfer Δ > 0 from country i to country j, such that positive contributions t_i^Δ and t_j^Δ continue to result in equilibrium. Then total equilibrium contributions increase: T ^Δ > T.

Next, we investigate the issue of burden sharing in this context. What can be said with respect to the contributions of the various countries in relation to the economic parameters characterizing these countries?

Let T_i := k_i t_i denote the total contribution of country i, i ∈ N, in equilibrium. Then we obtain the following result regarding burden sharing (the proof is again given in the Appendix):

Theorem 3.1. For any two countries i and j in N relative burden sharing is related to awareness and “real” GDP per capita, w^{^}_i , resp. “real” GDP, W^{^}_i , in the following way:

t_i / w^{^}_i ≤ t_i / w^{^}_i ⇔ α_i w^{^}_i ≤ α_j w^{^}_j, or T_i / W^{^}_i ≤ T_i / W^{^}_j ⇔ α_i w^{^}_i ≤ α_j w^{^}_j (7)

This theorem shows first of all that proportional burden sharing (with respect to “real” income) for mitigating climate change is the exception, although it could provide a basis for an equitable or fair allocation in this context. Moulin analyzes theoretical issues of equitable allocations (cf. Moulin 1987), whereas Horstmann and Scholz (2011) address more practical aspects in the context of mitigating climate change.

Moreover, a proportionally higher share of the burden arises not only from a higher GDP or GDP per capita. The effect of “awareness” has to be taken into account. Thus, disentangling the influences of awareness and GDP per capita, it is possible that despite of a high GDP per capita, a proportionally lower share of the burden results from a low level of awareness.

The following corollary reformulates this result and relates it to the issue of equity, and the empirical context of Fig. 2 below. Its proof follows immediately from Theorem 3.1.

Corollary 3.1 (EKC).

Assume w. l. o. g. that w₁ ≤ … ≤ w_n. Then β₁t₁ / w₁ ≤ … ≤ β_nt_n / w_n, and β₁T₁ / W₁ ≤ … ≤ β_nT_n / W_n, if and only if β₁α₁w₁ ≤ … ≤ β_n α_n w_n.

Thus, the question, whether the share of GDP for expenses on renewable energies increases with GDP, is in particular dependent on the level of awareness of climate change. Obviously, the combination of the two variables plays a role: a lower awareness can be compensated through a higher real GDP per capita and vice versa. What can be said empirically about the structures of both α and w^{^} depending on GDP per capita?

The following section addresses these empirical issues in a basic and preliminary context. In particular, the question of an increasing function α (w), will be investigated. These results depend critically on empirical values for LCOE, the levelized costs of renewable energy production, for which we have only rough estimates.

We investigate and evaluate the share of “Renewable Energy Consumption” in various countries as an indicator of awareness regarding climate change. The first step consists in determining roughly the values of the cost factors β_i for countries i ∈ N in consideration, i.e., the LCOE of 1 kWh of electricity generated by means of renewable sources.

There is a substantial range of costs, depending on the situation of a particular country (cost of equipment, labor costs, climatic situation, etc.) and on the combination of technologies to generate electricity from renewable sources (cf. Figure 3 in WEC, 2013, p. 11).5 However, precise data on average LCOE in a particular country seem not to be available so far. In order to complete our empirical analysis, we thus have to make some rudimentary assumptions regarding the values of the β_i, i ∈ N.

In order to approximate the scarcely available empirical data, we define the function β (w) in the following simple way: β (w):= –0.1 + 0.0000066 w. This leads to LCOE of approximately 0.07 US-$ per kWh for a country with a GDP per capita (PPP) of 25,000 US-$ (Portugal), of approximately 0.19 US-$ for a country with a GDP per capita (PPP) of 45,000 US-$ (Germany), and of approximately 0,26 US-$ per kWh for a country with a GDP per capita (PPP) of 55,000 US-$ (US). β (w) then connects these pairs of coordinates by a straight line. The values for the β_i, i ∈ N, correspond approximately to some estimates provided in the literature, although precise data are required for a sound analysis. Observe that with this specification of β (w) the function w^{^}(w) := w /β (w) decreases with w increasing.

If we look again at current efforts of various industrialized countries to mitigate climate change by making use of renewable energy sources (without hydroelectricity), we obtain the situation indicated in Fig. 2, with the LCOE given by the above function β (w).

Again, there is no clear structure detectable. In particular the share of expenses of GDP on renewable energy consumption does not increase with GDP per capita (cf. also Fig. 1). Consequently, in view of Corollary 3.1, β_i α_i w_i cannot increase with GDP pc = w_i of these countries.

For the formal background of the empirical investigations, we consider the variables α_i as functions of the variables (T_j, k_j, w_j, β_j)_j∈N. Instead of solving the expressions for the equilibrium contributions for the parameters α_i, i ∈ N, we make directly use of the first order conditions for the interior Nash equilibrium:

k₁t₁ + … + (k_i + α_i) t_i + … + k_n t_n = α_i w^{^}_i for i ∈ N (8)

with “real” GDP per capita w^{^}(w):= w_i /β_i. These equations can be rewritten as follows: T = α_i (w^{^}_i – t_i) with total contributions towards the provision of the public good, i.e., total spending on renewable energy sources, of the member states, given by T = k₁t₁ + … + k_n t_n. Consequently, the values of the α_i, i ∈ N, result immediately from the observable, or computable parameters T, w^{^}_i and t_i:

α_i = T / w^{^}_i – t_i for i ∈ N (9)

The first analysis investigates climate-sensitive behavior of Annex II Parties to the Kyoto Protocol, industrialized countries in the OECD with self-commitments to reduce greenhouse gas emissions.

The United Nations Framework Convention on Climate Change (UNFCCC), resulting from the “Earth Summit” in Rio de Janeiro in 1992, divides countries into three main groups according to differing commitments:6

Table 1 presents the results for the values of GDP per capita (2013 US-$, ppp) and “environmental awareness” for selected Annex II Parties. The countries are ranked according to the values of environmental awareness. The selected countries are somewhat comparable, although not identical, regarding their economic development.

Countries with a supposedly high environmental awareness (Norway, Switzerland, US, Sweden) lead this list of Annex II countries. The only surprise is that the Southern European countries Italy, Spain, Portugal and Greece with a lot of sunshine appear towards the lower end of the ranking.

Observe that in this table we do not include consumption of electricity from hydro power plants, as the availability of hydroelectricity is largely dependent on appropriate geographical conditions. Nevertheless, the abundance of hydroelectricity in some countries might affect consumption of electricity from the other renewable sources. This might in turn have an effect on the empirical value of “awareness”, not taken into account in the above estimations (cf., however, Fig. 3).

One of the prominent questions arising in this context refers to the relationship between awareness of climate change and GDP per capita. More precisely, this is the question, whether this relationship reveals aspects of an EKC (cf. Stern, 2004; Huang et al. 2008; Kaika and Zervas, 2013a, 2013b; Dong et al., 2016). Fig. 3 presents the result based on the LCOE function β (w):= –0.1 + 0.0000066 w for renewable energies with and without hydropower.

There is, thus, a rather clear tendency for higher environmental awareness to be associated with a higher GDP per capita. This result is in favor of the upward sloping part of a classical EKC, implying that activities for mitigating climate change increase with economic wealth. However, the results do not point to a classical EKC.

We know from Corollary 3.1 that the behavior of β_i α_i w_i determines the behavior of β_i t_i /w_i = β_i T_i /W_i = β_i T_i /GDP_i, i ∈ N. In this sense, the empirical curve α (w) from Fig. 3 and the downward sloping function w^{^}(w) “explain” the structure of Fig. 2. Although awareness shows an increasing trend, this positive effect is more than compensated through the negative income effect resulting from the decreasing “real” income.