For a monopolist facing a downward sloping function P = f(Q) and a standard total revenue function TR = R (Q), the cost function is given by TC = C(Q) + F. Two alternative taxes can be separately imposed on the firm: namely a percentage tax on sales revenue at the rate of s (so that after tax total revenue becomes (1-s) R (Q) ) or an excise tax of $t per unit (so that after tax cost becomes C (Q) + t Q +F). The tax rates are designed in such a way that for a profit maximizing firm, the after-tax optimal output is the same under the two alternative tax structures.

Prove that in this situation the revenue tax can generate more tax revenue than the excise tax. Note: You need to work with general functions as described above and not any specific example.

(Hint: First prove that for such tax rates: sMR = t , where MR is the after-tax MR -the common value of MR at the post tax optimal output, which by design is the same under the two taxes. Then apply the relationship between P and MR.)