Consider the case when the cost function satisfies C_xe(•)=0.
This can arise for different reasons. Consider the following two possibilities:

(a) Suppose production and abatement costs (AC) are separable, so that C(x,e)=Č(x)+AC(e). Show that in this case an emissions tax does not affect the output price.

(b) Suppose that C_xx(•)≥0 and abatement costs are linear. For example, C(x,e)=c•x+β(δ•x−e). Show that the firm’s optimal choice for emissions is a corner solution. In particular, show that for τ<β the firm does not reduce any emissions, and for τ>β it reduces emissions to zero.