Derivation of (13.3). Since non-signatories will play Pollute, the aggregate payoff of signatories, ∏s, can be written as:(13A.1)
Maximization of ∏s with respect to zi requires(13A.2)for i = 1, 2. Assuming for simplicity that c/bi is a non-integer so that (13A.2) holds with strict inequality, the solution requires that all signatories of type i play either Abate or Pollute. There are three kinds of equilibria: either Z1* = Z2* = 0 or Z1* = 0, Z2* = k2 or Z1* = k1, Z2* = k2. Note that Z1* = k1, Z2* = 0, though feasible, cannot be an equilibrium. This means that, in an agreement consisting of both types of country, if it is optimal for type 1 signatories to play Abate then it must be optimal for type 2 signatories to play Abate.
Derivation of the Conditions for Equilibria Consisting of Both Types of Country Based on the Chander-Tulkens Cost-sharing Rule. Suppose to begin that α1k1 + k2 > c/b1. Then a type 1 country won't accede if α1(b1k1 + b2k2) > −cα1(k1 + 1 + k2)/ [α1(k1 + 1) + k2] + α1[b1(k1 + 1) + b2k2] or c/b1 > [α1(k1 + 1) + k2]/(k1 + 1 + k2), which holds by the model's assumptions. Similarly, a type 2 country won't accede, since c/b2 > (α1k1 + k2 + 1)/(k1 + k2 + 1).
Upon a withdrawal from an equilibrium agreement, two possibilities must be considered. Either all the type 1 signatories must play Pollute or all the signatories of both types must do so. If neither of these conditions were satisfied, then a withdrawal from the agreement would always be individually rational.
Consider first the former possibility. For a withdrawal by a type 1 signatory, c/b1 > α1(k1 − 1) + k2 > c/b2 and for a withdrawal by a type 2 signatory, c/b1 > α1k1 + k2 − 1 > c/b2. A unilateral withdrawal by a country of either type will then cause all the remaining type 1 signatories to play Pollute if c/b1 + α1 > α1k1 + k2 > c/b2 + 1. Given this, a type 1 signatory would not withdraw if α1k1 + k2 > c(k1 + k2)/(b1k1); and a type 2 signatory would not withdraw if α1k1 + k2 > c(k1 + k2)/(b1k1 + k2). Taken together, these conditions yield:(13A.3a)
Now suppose that a withdrawal impels all signatories to play Pollute. Then we have c/b2 + α1 > α1k1 + k2. Withdrawal by a type 1 signatory will therefore be irrational if α1k1 + k2 > c(k1 + k2)/(b1k1 + b2k2). Since the RHS of this inequality cannot exceed c/b1, and since we are considering the case where α1k1 + k2 > c/b1, we know that a