Multilateralism, Bilateralism and Regime Design

Each member  ∈ [1 ] chooses her level of  ≥ 0 to maximize her utility  = √− P  6= −  a function that is twice-differentiable and concave. Assuming  ≥ 0 to be the Lagrangian parameters, the optimal level of   #   satisfies the necessary and sufficient first-order conditions  1 2 − 1 2  −  +  = 0 and the Kuhn-Tucker conditions  = 0 for any  ∈ [1  ], thus forming a system of 2 equations and 2 variables ( and ). There is no solution possible in which, for any member  ∈ [1  ],   0 because it would imply # = 0 making the corresponding first-order condition indeterminate. Therefore, the only possible determinate solution has  = 0 and # = ¡  2 ¢2 for all  ∈ [1 ] 