Question

Show that a profit-maximizing monopolist's output is unaffected by a propor- tional profit tax, but is reduced by a tax of $t per unit of output. Explain these results

Answer 1

Demand curve of monopolist: P = a - bQ [P: Price, Q: Output]

Total cost of monopolist: C = c + dQ

Total revenue TR = P x Q = Q x (a - bQ) = aQ - bQ2

Marginal revenue, MR = dTR / dQ = a - 2bQ

Marginal cost, MC = dC / dQ = d

A monopolist will maximize profit by equating MR with MC:

a - 2bQ = d

2bQ = d - a

Q = (d - a) / 2b

Since profit-maximizing output is independent of the price, therefore a profit tax will not change its output.

But a tax of $1 per unit of output will increase the marginal cost, hence the revised MC will be:

MC = d + 1

Equating MR with MC,

a - 2bQ = d + 1

2bQ = a - d - 1

Q = (a - d - 1) / 2b < (a - d) / 2b [The before-tax output]

Therefore, output decreases with imposition of output tax.

This difference is because, unlike in the case of a profit tax, the output tax is a tax imposed before the profit figure is derived. Such a tax increases the cost of production, so monopolist sells less.