Question

Monopoly

Consider a monopoly facing an inverse demand function P(q) = 9 − q and having a cost
function C(q) = q.

(a) Find the profit maximizing output and price, and calculate the monopolist’s profits.

(b) Now, suppose the government imposes a per unit tax t = 2 to the monopoly. Find the
new price, output and profits. Discuss the impact of that tax.

Answer 1

a) Inverse demand function for the monopolist is P(q)=9-q

and cost function C(q)=q

Then, MR = d/dq(TR) = d/dq(P*q) = d/dq(9q-q²) = 9-2q

and MC =d/dq(C)= d/dq(q) = 1

Then , for profit maximization, MR=MC

or, 9-2q=1

or, 2q=8

or, q=4

and P=9-q = 9-4 = $5

Profit of the monopolist = TR-TC = P*q - C = 5*4 - 4 = $16

b) With the per unit tax imposed on the monopolist,the effective inverse demand function P'(q)=9-q'-T = 9-q'-2 = 7-q'

Then, MR=d/dq'(7q-q'²) = 7-2q'

Now, for equilibrium, MR=MC

or, 7-2q'=1

or, 2q'=6

or, q'=3

New price P'=9-q' = $(9-3) = $6

New profit = TR-TC = 6*3-3 = $15

Thus, taxes on monopolists decreases its revenue and thereby profit. Taxes also increases price and reduces output.