An industry has many buyers and one seller. Market demand for the good is given by
p = A − Q
where p is price, Q is quantity, and A > 0. The monopolist's total cost function is
TC(Q) = B Q + Q2
where B > 0.
Suppose in addition to the cost of production, the monopolist faces a tax of t per unit sold, meaning that if Q units are produced and sold, t Q (that is, t multiplied by Q) is paid to the government. Assume t >= 0. a. Write the monopolist's total cost and marginal cost functions, inclusive of the tax.
b. Compute the monopolist's optimal choice of quantity as a function of t (and parameters A and B). (You may ignore the possibility that the monopolist shuts down.)
c. Express the government's total revenue from the tax (that is, the amount paid by the monopolist to the government), as a function of t (and parameters A and B).
d. Calculate the government's choice of the tax rate t that maximises its revenue from the tax (as a function of parameters A and B).