# application of differentiation in economics

#### helloprajna

A monopolist produces x units of a commodity at a total cost of ax^2+bx+c and the demand law for the same is given by
p= Q-Rx, where Q and R constants.
A tax of Rs.K per unit is levied by the government. The manufacturer adds the tax to his cost.

(a) find the price and the output both before and after tax.
(b) show that the tax brings in maximum return when K=(Q-b)/2

I could solve the 1st question i.e. (a). But I don't know how to solve (b). Please help.

#### CRGreathouse

Forum Staff
Tax is maximized when the derivative of tax revenue is 0. First solve for tax revenue, then differentiate. Next set this equal to zero and solve. Finally, substitute back into the original.

#### helloprajna

CRGreathouse said:
Tax is maximized when the derivative of tax revenue is 0. First solve for tax revenue, then differentiate. Next set this equal to zero and solve. Finally, substitute back into the original.
Does that mean 'maximum return' refers to 'maximum total tax revenue' here???

Forum Staff
Yes.

#### helloprajna

CRGreathouse said:
Thank you so much....