Question

A Monopoly faces market demand given by Q = 100 – 2P, where Q stands for quantity and P for price. Total cost function is given by C (Q) = 10Q. Find the profit maximizing price and quantity and the resulting profit to the monopoly. Also show that the equilibrium price adheres to the optimal markup rule based on demand elasticity.

Answer 1

Derive Total Revenue (TR):

First solve for P:

Q = 100 – 2P

2P = 100 – 0.5Q

P = 50 – 0.5Q

TR = P*Q

TR = (50 – 0.5Q) Q

TR = 50Q – 0.5Q²

Derive marginal revenue:

MR = derivative of TR with respect to Q

$\frac{\partial TR}{\partial Q }$ = 50 – Q

MR = 50 – Q

Compute the profit maximizing output by setting MR = MC:

MC = derivative of TC with respect to Q

TC = 10Q

MC =$\frac{\partial TC}{\partial Q }$ = 10

MR = MC

50 – Q = 10

Q = 50 - 40

Q = 40

Profit maximizing output for the monopoly = 40

Profit maximizing price = Substituting Q in the demand function

P = 50 – 0.5Q

P = 50 – 0.5(40)

P = 50 – 20

P = 30

Profit maximizing price for the monopoly = 30

 

Profit = TR – TC

= (50(40) – 0.5(40²)) – 10(40)

= (2000 – 800) – 400

= 1200 – 400

= 800

Profit for the monopoly = 800

Equilibrium price adheres to the optimal markup rule-based on-demand elasticity:

First, calculate the elasticity at P = $30, Q = 40

E = $\frac{\partial Q}{\partial P} (\frac{P}{Q} ) = 2[\frac{30}{40}] = 1.5$

Then plug 1.5 into the markup rule

P = $\frac{MC}{[1 - \frac{1}{E}] } = \frac{10}{[1 - \frac{1}{1.5}] } = \frac{10}{[ \frac{1}{3}] } = 30$