There are three equations determining consumption, emissions and environmental quality. Total differentiation yields:
and by applying Cramer's rule, we obtain
The numerator of the fractions on the right-hand side, i.e. the determinant of the matrix in equation (3.A1) is negative. Among the environmental-impact parameters, a has been taken as an example. The impact of the other parameters follows by analogy. It follows that the effects on the autarky rate of return on capital are
The marginal productivity of capital is increased if the impact parameter is raised. The effect of the emission tax rate is ambiguous.
Totally differentiating equations (3.2), (3.3), and (3.4) yields