Proposition 6.9 is proved by comparing the slopes of the domestic firm's iso-profit curve and the foreign firm's reaction curve in the Nash equilibrium with strategic choice of emission taxes. We assume that the marginal environmental damage is fixed, i.e. u′ is constant. The profits of the domestic firm can be expressed as
π(te,s) = π(sau′) + c + ̃(sau′) - c + ̃(te,s),
where te,s is the strategic tax rate, given by equation (6.24). The profits of a firm subject to this taxation equal the profits of a firm subject to the Pigouvian tax rate, sau′, plus the subsidy which is the cost with Pigouvian taxation minus the cost with strategic taxation. The other arguments of the functions have been omitted for convenience. The profit maximum is determined by:
πq(sau′) + c + ̃q(sau′) - c + ̃q(te,s) = 0. (6.A1)
Using a Taylor series approximation, one obtains
πq(sau′) - (te,s - sau′)c + ̃qt(te,s) - ζ = 0. (6.A2)
ζ is a positive term for the correction of the error which is due to the approximation of the non-linear concave function by a linear one. For te,s - sau′, one can use equation (6.24) and after some rearranging of terms, one arrives at(6.A3)
where the arguments of the functions have been omitted for convenience. It follows from equation (6.22) that the slope of the foreign firm's reaction curve can be represented by R + ̆q = Qt/qt < 0. Moreover, πq/πQ is the slope of the iso-profit curve (for a firm subject to Pigouvian taxes). The numerator of the first term on the right-hand side equals one if there is no transfrontier pollution. In this case the isoprofit curve is flatter than the foreign firm's reaction curve. In the case of substantial transfrontier pollution the numerator is greater than one and the iso-profit curve may be steeper. This proves proposition 6.9. If the environmental-damage function is non-linear and the marginal damage is increasing, then the first effect is reinforced and a scenario where the strategic Nash equilibrium is located left of the Stackelberg point becomes more likely.