Unilateral Actions the Case of International Environmental Problems

B_i (q) measures the benefits to country i from total reduction, q. On the other
hand, costs of controlling emissions only depend on own reductions, qi, and are
measured by Ci (qi) . We make the standard assumptions on the functions that
B_i^'(q) > 0 and Bi^''(q) < 0 while C_i^'(q_i) > 0 and C^'i^'(qi) > 0 .³ Hence, the net-benefit
to country i from own and total reduction efforts amounts to:

NBi =Bi(q)-Ci(qi), i=1,2

Given the assumptions, the net benefit functions are strictly concave in qi. If
each country behaves non co-operatively, it maximises its own net-benefit func-
tion with respect to its own reductions, qi, considering only damage in its own
country but not the damage it causes in other countries, or alternatively, not
considering the public good character of own reduction. The first order condi-
tion for a country to maximize its net benefit in a non-cooperative setting is ob-
tained by differentiating (1) with respect to own reductions, taking the other
countries’ reductions, q-i as given:

NBi^' =Bi^'(q)-Ci^'(qi)=0

Define the best reply function, or the reaction function of country i, as the func-
tion that relates the optimal choice of country i to the choice of reduction of the
other countries, qi = qi(q-i) . The slope of this function is determined as
^dqi(^q-i) = `b `` .⁴ Since B"(q) < 0 Ci(q_i) > 0, we have that -1 < ^dqi(^q-i) < 0.
_{∂q-i       Ci}'' _{- Bi}''                  i                i i                                           _∂q-i

Moreover, given the assumptions on the benefit and cost functions, there exists
a unique (and interior) Nash-equilibrium given by q^nc = {q1^nc,q2^nc,,,qn^nc} at the

3 For a more detailed discussion of these assumptions, see e.g. Finus (2001).

4 Conditions that guarantee a unique (and interior) Nash equilibrium are given in Finus (2001,
chapter 9).

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