Corresponding author: Shlomo Weber ( sweber@mail.smu.edu ) © 2019 Nonprofit partnership “Voprosy Ekonomiki”.
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Citation:
Weber S, Wiesmeth H (2018) Environmental awareness: The case of climate change. Russian Journal of Economics 4(4): 328345. https://doi.org/10.3897/j.ruje.4.33619

The extent of provision of a public good often relies on social awareness and public support for it. This applies, in particular, to global reduction of greenhouse gases and its relevance for mitigating climate change. We examine the concept of “public awareness” by introducing a formal model that analyzes efforts to mitigate climate change in a setting with heterogeneous countries. In the theoretical part we examine the Nash equilibrium of the contribution game. The effects of awareness and economic parameters on mitigation efforts can be disentangled, raising the possibility of linking awareness of climate change with economic wealth. The second part provides some empirical observations and offers the rankings of countries regarding awareness for climate change, as well as an empirical relationship between awareness and economic wealth.
environmental awareness, climate change, public goods, Kyoto Protocol, renewable energies, regional economics, diversity.
In December 2015, the parties to the Kyoto Protocol reached an agreement for reducing anthropogenic greenhouse gas emissions — after years of negotiations. The outcome of the Conference of the Parties (COP 21) in Paris is generally considered positive, albeit insufficient regarding the ambitious goal of limiting global warming to 1.5 degrees. There seems to be some agreement among the scientific community that further actions are needed, in particular after 2030.
The critical issues, impeding further reaching commitments so far, refer to sharing the cost of this global environmental commodity among highly diverse countries. In this context, Article 3.1 of the United Nations Framework Convention on Climate Change (UNFCCC) requires that the mitigation and adaptation efforts be shared between the parties “on the basis of equity and in accordance with their common but differentiated responsibilities and respective capabilities”. If, in this context, we look at current efforts of various industrialized countries to mitigate climate change by making use of renewable energy sources (with and without hydroelectricity), we obtain the situation indicated in Fig.
Share of energy consumption from renewable sources (with and without hydropower) depending on GDP (2013 thousand US$, pc, ppp) of various industrialized countries.
Source: Authors’ calculations with data from http://data.worldbank.org/ and http://www.bp.com/
There is no clear structure detectable, not with respect to usual concepts of equity (cf.
Of course, there are other reasons in the various countries to promote renewable energies, the development of new technologies, for example, or to gain more independence from the import of fossil fuels. However, climate change mitigation continues to play a decisive role, as emphasized in, among other sources, the EUDirective on the promotion of the use of energy from renewable sources (cf.
This paper therefore investigates effects of diversity on efforts to mitigate global warming. In particular, what role does “awareness” of climate change play in combination with economic variables? Should we expect a positive relationship between awareness and economic wealth? What can we learn about “awareness” in our model from empirical data?
As a result of the Lima Climate Change Conference in December 2014, the parties to the Kyoto Protocol were invited “to communicate their intended nationally determined contributions well in advance of the twentyfirst session of the Conference of the Parties in a manner that facilitates the clarity, transparency and understanding of the intended nationally determined contributions” (cf.
The following section reviews the relevant literature, mainly with respect to climate change and reducing greenhouse gas emissions. Thereafter, we introduce the model with diverse countries. Observable diversity refers mainly to GDP per capita and costs of renewable energy consumption. “Awareness” refers to not directly observable characteristics of a country. The Nash equilibrium, resulting from the interaction of the countries, allows some insight into the effects of diversity. In particular, properties of equilibrium burden sharing in its relation to equity and also in relation to observable characteristics can be analyzed. The then following section is dedicated to some empirical investigations. In particular, awareness will be estimated from observable data, allowing some conclusions concerning the dependence on observable characteristics. Some final remarks conclude the paper.
The literature on the voluntary provision of public goods is abundant, covering nearly all aspects of theoretical and practical relevance (cf., for example, the seminal works by
The paper by
These basic models on the provision of public goods have been adapted to the environmental context in various ways (cf. the papers mentioned below).
Interest in the concept of “environmental awareness” or “environmental consciousness” originated with the ecological movement in the 1960s. According to
“Personal value orientation” as precursor of sustainable behavior was considered in a further stream of research followed by “cultural values”, which have been investigated for the last ten years or so (cf. again
It seems to be plausible to assume that environmental commodities are characterized by a relatively highincome elasticity, at least in industrialized and newly industrialized countries. Consequently, demand for these commodities should rise, and environmental pollution should be reduced with real GDP per capita further increasing. The resulting functional relationship between the level of pollution and GDP per capita points to aspects of an EKC. If such a relationship holds for a multitude of environmental issues, an increasing economic welfare would gradually help to solve local environmental problems. Regarding climate change, global greenhouse gas emissions will nonetheless continue to increase as long as sufficiently many countries raise their emissions.
In a local context, Grossman and Krueger “find no evidence that environmental quality deteriorates steadily with economic growth. Rather, for most indicators, economic growth brings an initial phase of deterioration followed by a subsequent phase of improvement. The turning points for the different pollutants vary, but in most cases, they come before a country reaches a per capita income of $8000”. Their study uses urban air pollution, the state of the oxygen regime in river basins, fecal contamination of river basins, and contamination of river basins by heavy metals as indicators (
Similarly, in a context of water pollution in countries in Central and Eastern Europe with the indicator “biological oxygen demand” (BOD),
However, despite these promising examples, according to
Share of Expenses on Renewable Energies depending on GDP pc (2013 thousand US$, ppp) of various industrialized countries.
Source: Authors’ calculations using β (w) with data from http://data.worldbank.org/ and http://www.bp.com/
Awareness depending on GDP pc (2013 thous. US$, ppp) of the Annex II countries listed in Table
Source: Authors’ calculations with data from http://data.worldbank.org/ and http://www.bp.com/
In the last years, more and more advanced econometric techniques were employed to investigate existence or nonexistence of the EKC.
Recently, other empirical investigations revealed interesting aspects of the willingness to pay for climate actions. In this context
Awareness regarding climate change has also been addressed in various publications.
So far, the review of the literature leaves us with a somewhat unclear picture. On the one hand we have empirical examples of an EKC, on the other hand we have the investigations of
The above considerations show that there is enough room and also a certain necessity for explicitly introducing “awareness’’ into an economic model to mitigate global warming — to capture noneconomic aspects of diversity among the countries. The first subsection presents the main assumptions of the model, emphasizing diversity.
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
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^{αi} 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) ∈ IR^{2}_{++}. 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
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.
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} (t_{1}^{1},…, t_{1}^{k}^{1}; …; t_{n}^{1},…, t_{n}^{kn}) :=
= (w_{i} – β_{i} t_{i}^{1}) ∙ (t_{1}^{1} + … + t_{1}^{k}^{1} + … + t_{n}^{1} + … + t_{n}^{kn})^{αi} =
= x_{i}^{1}y^{αi} = u_{i} (x_{i}^{1}, y) (2)
We make the following assumption with respect to the utilitymaximizing 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.
We then obtain the following first order condition for individual 1 in country i ∈ N:
β_{i} (t_{1}^{1} + … + t_{1}^{k}^{1} + … + t_{n}^{1} + … + t_{n}^{kn}) = α_{i} (w_{i} – β_{i} t_{i}^{1}) (3)
As the lefthand sides of these first order conditions for the individuals of country i are identical, the righthand sides must be identical, too, resulting in identical equilibrium contributions of all agents of this country. Thus t_{i}^{1} = … = t_{i}^{ki} =: t_{i} in equilibrium for each i ∈ N. Consequently, the first order conditions for i ∈ N can be rewritten as follows:
k _{1}t_{1} + … + (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, egalitarianequivalent allocations or core allocations (cf.
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 firstorder conditions for an interior solution are given by: k_{1}t_{1} + … + (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 = (t_{1}, …, t_{n}) ∈ IR^{2}_{++} for the Nashequilibrium. The proof of Proposition 3.1 is provided in the Appendix.
Proposition 3.1. First of all, the solution t = (t_{1}, …, t_{n}) ∈ IR^{2}_{++} 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, t_{1} is given by:
t _{1} = [k_{2} (α_{1}α^{^}_{2} … α_{n}) w^{^}_{1} – k_{2} (α^{^}_{1}α_{2} … α_{n}) w^{^}_{2} + … + k_{n} (α_{1}α_{2} … α^{^}_{n}) w^{^}_{1} –
– k_{n} (α^{^}_{1}α_{2} … α_{n}) w^{^}_{n} + (α_{1} … α_{n}) w^{^}_{n}] /
/ [k_{1} (α^{^}_{1}α_{2} … α_{n}) + … + k_{n} (α_{1} … α^{^}_{n}) + (α_{1}α_{2} … α_{n})] (5)
with α^{^}_{i} meaning that this factor has to be replaced by 1. By using α^{–}_{i} := (α_{1} … α^{^}_{i} … α_{n}) and α^{–} := (α_{1}α_{2} … α_{n}) , we can simplify this expression in the following way:
Again, we assume that an interior solution with t_{1}> 0, i ∈ N, exists for the given constellation of the parameters.
Returning to individual equilibrium contributions t_{i}^{n} and total equilibrium contributions T = k_{1}t_{1} + … + 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
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.
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.
Corollary 3.1 (EKC).
Assume w. l. o. g. that w_{1} ≤ … ≤ w_{n}. Then β_{1}t_{1} / w_{1} ≤ … ≤ β_{n}t_{n} / w_{n}, and β_{1}T_{1} / W_{1} ≤ … ≤ β_{n}T_{n} / W_{n}, if and only if β_{1}α_{1}w_{1} ≤ … ≤ 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.
With this empirical analysis we want to derive first estimates for the awareness parameter — given the ordinal specification of our utility functions — from the observable data on population, GDP, and renewable energy consumption and the herewith associated cost estimates. In addition, in view of Corollary 3.1 the goal is to understand the empirical relationship between awareness and GDP per capita. As already indicated in the introductory section, we focus on countries, which are parties to the Kyoto Protocol.
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
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.
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.
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:
k _{1}t_{1} + … + (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 = k_{1}t_{1} + … + 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 climatesensitive behavior of Annex II Parties to the Kyoto Protocol, industrialized countries in the OECD with selfcommitments 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:
Table
Ranking of Annex II Parties regarding “awareness”.
Country  GDPpc  α_{i}  Country  GDPpc  α_{i}  
Norway  66,520  0.407  Switzerland  56,580  0.386  
US  53,960  0.380  Sweden  44,760  0.352  
Denmark  44,460  0.351  Germany  44,540  0.350  
Austria  43,840  0.346  Netherlands  43,210  0.343  
Canada  42,610  0.340  Australia  42,540  0.340  
Belgium  40,280  0.330  Finland  38,480  0.322  
Japan  37,630  0.315  France  37,580  0.314  
UK  35,760  0.304  Ireland  35,090  0.300  
Italy  34,100  0.293  Spain  31,850  0.277  
New Zealand  30,750  0.269  Greece  25,630  0.215  
Portugal  25,360  0.213 
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.
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.
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.
The theoretical part of this paper analyzes the interaction of the agents of various countries regarding efforts to mitigate climate change. These efforts are measured by renewable energy consumption and the interaction is governed by the Nash mechanism. The results demonstrate the influence of “awareness”, in addition to the economic variable “GDP per capita” on burden sharing.
The empirical part of the paper makes use of the firstorder conditions to allow an explicit computation of the awareness parameters for various countries. The results are dependent on the levelized costs of energy from renewable sources, for which there are only more or less rough estimates. The estimates of LCOE applied here lead to aspects of an empirical EKC for awareness of climate change.
Future research in this context could focus on this more or less latent variable “awareness”. Which parameters influence awareness, and how could awareness be raised in order to accelerate efforts to mitigating climate change. However, it is also necessary to improve the empirical analysis through a more careful estimate of the levelized costs of energy from renewable sources. So far, the estimates available in the literature are rather imprecise and incomplete.
Part of the paper was written while Hans Wiesmeth was visiting SMU in Dallas and NES in Moscow, and Shlomo Weber was visiting TU Dresden. Financial support from the Department of Economics at SMU and LISOMO at NES, and the Faculty of Economics at TU Dresden are gratefully acknowledged.
Shlomo Weber is grateful to the “Domodedovo” group of companies for financial support of the NES Center for the Study of Diversity and Social Interactions in 2018–2019. This paper is the continuation of the research conducted under auspices of the grant No. 14.U04.31.0002 of the Ministry of Education and Science of the Russian Federation administered through NES CSDSI in 2013–2017. The research work of Hans Wiesmeth was gratefully supported by Act 211 Government of the Russian Federation, contract No. 02.A03.21.0006.
The authors are grateful to an anonymous referee, who pointed to various issues regarding the empirical analysis and future research.
Proof of Proposition 3.1: For the proof we note that the system of first order conditions is symmetric in the sense that symmetrically exchanging indices leads from one equation to the other ones. Therefore, this property is retained for the solutions. Next, we show that the first one of these first order conditions is fulfilled by plugging in the above values of t_{i}. This equation reads:
(k_{1} + α_{1}) t_{i} + k_{2}t_{2} + … + k_{n} t_{n} = α_{1}w^{^}_{1} (A1)
From straightforward calculations we immediately obtain the following expression for the total quantity of the public commodity T = k_{1}t_{1} + … + k_{n} t_{n} provided in equilibrium:
Thus, it remains to show: T + α_{1}t_{1} = α_{1}w^{^}_{1}. Canceling out α_{1} and rearranging the terms by means of the above formula we obtain:
( (T /α_{1}) + t_{1}) (k_{1}α^{–}_{1} + … + k_{n} α^{–}_{n} + α^{–}) = k_{1}α^{–}_{1}w^{^}_{1} + … + k_{n} α^{–}_{1}w^{^}_{n} +
+ k_{2}α^{–}_{2}w^{^}_{1} – k_{2}α^{–}_{1}w^{^}_{2} + k_{3}α^{–}_{3}w^{^}_{1} + … + k_{n} α^{–}_{n}w^{^}_{1} – k_{n} α^{–}_{1}w^{^}_{n} + α^{–}w^{^}_{1} =
= k_{1}α^{–}_{1}w^{^}_{1} + … + k_{n} α^{–}_{1}w^{^}_{n} + α^{–}w^{^}_{1} = w^{^}_{1}(k_{1}α^{–}_{1} + … + k_{n} α^{–}_{n} + α^{–}) (A3)
and the desired result follows. By making use of symmetry considerations, the other equations are also fulfilled with these values of the t_{i}, i ∈ N.
Proof of Result 3.2: For a total monetary transfer Δ from country i to country j each individual has to contribute the amount Δ/k_{i} and each individual of country j obtains the amount Δ/k_{j}. Consider then
from the proof of Proposition 3.1 above. We can similarly calculate T^{Δ} and investigate the difference to T by focussing on the relevant terms:
But this last expression is positive for β_{i} > β_{j}.
Proof of Theorem 3.1: In order to simplify the notation, we compare t_{1}w^{^}_{2} with t_{2}w^{^}_{1}. A straightforward calculation using real GDP and the nominators of the above terms yields:
t _{1}w^{^}_{2} ≤ t_{2}w^{^}_{1} ⇔
k _{2}α^{–}_{2}w^{^}_{1}w^{^}_{2} – k_{2}α^{–}_{1}w^{^}_{2}^{2} + … + k_{n} α^{–}_{n}w^{^}_{1}w^{^}_{2} – k_{n} α^{–}_{1}w^{^}_{2}w^{^}_{n} + α^{–}w^{^}_{1}w^{^}_{2} ≤
k _{1}α^{–}_{1}w^{^}_{1}w^{^}_{2} – k_{1}α^{–}_{2}w^{^}_{1}^{2} + … + k_{n} α^{–}_{n}w^{^}_{1}w^{^}_{2} – k_{n} α^{–}_{2}w^{^}_{1}w^{^}_{n} + α^{–}w^{^}_{1}w^{^}_{2} (A6)
Simplifying and substituting W^{^}_{i} for k_{i} w^{^}_{i}, i ∈ N, yields:
t _{1}w^{^}_{2} ≤ t_{2}w^{^}_{1} ⇔ (W^{^}_{1} + … + W^{^}_{n}) α^{–}_{2}w^{^}_{1} ≤ (W^{^}_{1} + … + W^{^}_{n}) α^{–}_{1}w^{^}_{2}
⇔ α^{–}_{1}w^{^}_{1} ≤ α^{–}_{2}w^{^}_{2} (A7)
thus, arriving at the desired result.