In this section, I drop the assumption of symmetry of countries and allow for individual parameters for each country. This means that the model is characterized by the number of countries n, the number of club members t and the parameter vector
π ∈ Rn+. (6.13)
I continue to assume γ ≡ 1. Proposition 9 shows results for global emissions EN in the cases of no cooperation or full cooperation among club members, when heterogeneity is present.
Proposition 9. Let
πT :=
∑
i∈T
πi, πR:=
∑
j∈R
πj. and i∈ T . Assume coalition S ⊂ T forms. Then
EN= EN0
πT+ πR+ 1 if S= {i} (6.14) and
EN= EN0
tπT+ πR+ 1if S= T. (6.15) These instances represent the case of no cooperation (formation of singleton coalitions) and the case of full cooperation among club members, respectively.
Proposition 9 shows that global emissions do not depend on the individual values of the damage parameter. Rather, they only depend on the parameter sum of both groups. This means that individual parameters πican be varied, while holding πT and πRconstant, without changing global emissions in the cooperative case. I use this property to study the impact of heterogeneity on the value of coalitions and therefore the existence of a stable allocation.
Due to the increased complexity of the case of asymmetric countries, con-ditions for a non-empty core cannot easily be calculated. Instead, I draw on Proposition 7 and focus on the set of singleton coalitions. Specifically, I check whether
i∈T
∑
v({i}) > v(T ), (6.16)
which is a sufficient condition for an emtpy core. The set of parameters for which condition (6.16) is not satisfied is calledP. To be clear, this means˜ that, while π /∈ ˜P is a sufficient condition for an empty core, π ∈ ˜P is
6.2 Theoretical results
only a necessary condition for a non-empty core. However, Proposition 7 showed that
P = P˜ (6.17)
in the symmetric case. Similar to Lemma 1, the elements ofP are deter-˜ mined by a quadratic function in πR:
Lemma 3. Let π ∈ Rn+. Then π ∈ ˜P if and only if
By design, condition (6.18) is equivalent to condition (6.4) in the symmetric case. Lemma 3 shows that heterogeneity of countries in R does not influence membership in ˜P, as the condition only depends on πR, the sum of parame-ters in R. The utility of club members is only influenced by the sum of emis-sions of outsiders, not their distribution. As the sum of emisemis-sions of outsiders does not depend on the distribution of parameters among outsider countries, the cooperation decision is not influenced by this distribution. However, it is clear that heterogeneity of countries in the club influences the core. Propo-sition 10 investigates this influence.
Proposition 10. Let π ∈ Rn+, l, h ∈ T with
πl≤πT
t , πh≥πT
t , (6.20)
and let d> 0 with
d≤ πl, d ≤ πT− πh. (6.21) Define ˜π ∈ Rn+by
π˜l= πl− d,
π˜h= πh+ d, (6.22)
π˜i= πi ∀i 6= l, i 6= h.
Then a( ˜π ) > a(π ).
π is designed such that ˜˜ πT = πT and ˜πR= πR, meaning that the transfor-mation "adds" heterogeneity without altering the sum of parameters in each group. The repeated application of Proposition 10 can construct any vector with these total group values. Therefore, the specific definition of ˜π repre-sents no loss of generality about the considered parameter vector.
Proposition 10 shows that the “addition” of heterogeneity in the club leads to an increase in a(π). The drivers behind this increase can be best visu-alized with a numeric example. Building on Figure 6.1, Figure 6.2 shows the impact of heterogeneity in the case of t = 3, n = 5. πT and πRare kept constant, while one country’s share of πTis increased. The remaining part of πT is divided equally between the other two club members. Focusing on the total net value a(π), we see that in the symmetric case (share of13) we have a(π) < 0, meaning that a stable agreement exists. However, a(π) increases with heterogeneity between the countries in T and a(π) is positive for shares larger than 0.55, resulting in an empty core of the corresponding game.
This increase of a(π) is driven by countries two and three, those with a de-creasing share of πT. Note that a2(π) = a3(π) and a(π) = ∑3i=1ai(π). As the ratio of damage cost and mitigation cost of countries two and three reduces, the gains of cooperation become smaller and a2(π) and a3(π) rise. The op-posite is true for country one: its ratio of damage cost and mitigation cost increases, leading to higher gains of cooperation and lower a1(π). However,
6.2 Theoretical results
this decrease is not big enough to offset the increases in a2(π) and a3(π).
Consequentially, a(π) increases with heterogeneity.
0.35 0.4 0.45 0.5 0.55 0.6 0.65
−8
−6
−4
−2 0 2 4 6 8 10 12
Share of πT of country one.
a(π)
a(π) a1(π) a2(π)=a3(π)
Figure 6.2: a(π) for varying heterogeneity between club members. πT = 1, πR = 0.2, t = 3, n= 5.
As the result of Proposition 10 holds for all πT and πR, the non-empty core condition (6.18) is violated for some parameter combinations previously in P after the execution of transformation (6.22). Therefore, the set ˜˜ P shrinks with increased heterogeneity, as visualized in Figure 6.3. It shows ˜P for dif-ferent distributions of πT upon the countries in T . These shares of countries are varied between subfigures only, meaning
πˆi= πi πT
, i ∈ T (6.23)
is constant for each subfigure.
ˆ
Figure 6.3: ˜P for different levels of heterogeneity in T.
The shrinking of ˜P with increased heterogeneity causes a different shape of the boundary of ˜P. While a high πRstill leads to less cooperation in T (as in the symmetric case), the same is true for a high πT, unlike the symmetric case. An intuitive explanation for this phenomenon is that, for asymmetric countries, the "absolute" differences in the parameters become more pro-nounced with higher πT, which reinforces the effect that more heterogeneity causes less cooperation. As the last subfigure shows, high diversity between countries in T can even lead to a game for which the core is empty for all combinations of πT and πR.
The results about cooperation of asymmetric countries align with the find-ings of Driessen et al. (2011), who show that, in a cooperative oligopoly game, higher heterogeneity of marginal costs decreases the size of the core.
This is also supported by the model of internal and external stability by Na