If the inverse demand function facing a monopoly is p(Q) and its cost function is C(Q), show the effect of a specific tax, on its profit-maximizing output. How does imposing affect its profit
The total profit is total revenue minus total cost and tax. π = TR − TC = P(Q)Q − tQ − C(Q) d π /dQ = P + (dp/dQ)Q – t – dc/dQ = 0 P + (dp/dQ)Q = t + dc/dQ.
After tax, output will be less and the monopoly will charge a higher price. To see the effect of the tax on profit we take the derivative of the profit function (using the envelope theorem) and we get:
d π /dt = −Q* < 0.
So, as a result of the specific tax, the profit decreases.