Question

A monopolist has a total cost given by 9Q^2+18Q+21 and faces a demand curve given by -4Q+120.

What is the profit-maximizing quantity the monopolist supplies the market? (Answer is 51/13, how did they get this?)

What is the profit-maximizing price the monopolist sets in this market? ((Answer is 1356/13, how did they get this?)

Suppose instead of a monopoly, the market is served by a bunch of perfectly competitive firms each with the same cost structure as the monopolist

What is the profit-maximizing quantity these firms supply the market? (Answer is 84, how did they get this?)

What is the profit-maximizing price in this market? (Answer is 9, how did they get this?)

What is the dead weight loss from having a monopoly versus a perfectly competitive market? (Answer is 3816, how did they get this)

Answer 1

Given - The Demand Curve is P = -4Q + 120___(1)

TC =  9Q^2+18Q+21___ (2)

a) We must first find the Total Revenue (TR). We know that TR is P*Q. Where P is Price and Q is Quantity,

TR = P*Q
from 1 we have

TR = (-4Q +120)*Q

= -4Q^2 + 120Q

TC =   9Q^2+18Q+21

To find the eq quantity we set MR = MC

We now find MR (Marginal Revenue) . It is the first order derivative of TR.

dTR/dQ = -8Q + 120

.It is given TC = 9Q^2+18Q+21

Hence MC = dTC/dQ = 18Q + 18

MR = MC

18Q + 18 = - 8Q + 120

26Q = 102

Q = 102/26 = 51/ 13

b) to see the price , we substitute Q = 51/13 in eqn 1

P = -4(51/13) +120.

= 1356 /13

c) In case of a perfectly competitive maket

P = MC ( P is price, M is marginal cost)

we have

P = 18Q + 18

From 2

-4Q+120 = 18Q + 18

22Q = 102

Q = 51/11

The profit maximising quantity is 51/11 ( not 84 as mentioned in the question)

d) The profit max price is P = -4(51/11) +120. = P = 101.454545455

e. To find dead weight loss we plot the graph for easy calculation as below

1356/ 13 is the price monopolists charge

and 51/13 is t he quantity they produce

101.45 is the price Perfect competition charges and they produce 51/11

The deadweight loss is the area of the dark triangle i.e 1/2 b * h

1/2 (1356/13 - 101.45)(51/11 - 51/13)

(Dead weight loss is 1/2 ( difference in price * difference in quantity)

= 1.01917697687