Unilateral Actions the Case of International Environmental Problems

10. Appendixes

Appendix 1

NBi =Bi(qi,qj)-C(qi), i=1,2.

B_i^'(q) > 0 and B_i^''(q) < 0 while C_i^'(q_i) > 0 and C_i^''(q_i) > 0 .

Comparative static. Suppose that a is some parameter that shifts the net benefit
function of country 1. The Nash equilibrium is described by the conditions:

^dNB1( q 1⁽ a )^, q 2( a )^, a ) = 0

^dq ι

^dNB 2( qι( a )^, q 2( a )) = 0

^dq 2

Differentiating these equations with respect to a gives us the system

'∂ ² NB1 ∂ ² NB1 '

^dq ι²^dq ι^dq 2

I ∂q 1 q 2 ∂q 2 J

'q ^] C ∂² NB, Ï
∂a = -
^dq 2 ^d^q'^sa

(Ba J ^ 0 ^j

Applying Cramer’s rule, we have

∂²NB₁ ∂²NB₁

^dq ι =

∂a

∂q₁ ∂a ∂q₁ ∂a

0 ∂² NB 2

∂q₂²

∂²NB₁ ∂²NB₁

∂q 1² ∂ q 1 ∂q 2

∂² NB ₂ ∂² NB ₂∂q₁ q₂ ∂q₂²

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