Answer to Question #180414 in Microeconomics for Darshana

Question #180414

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.


1
Expert's answer
2021-04-19T18:46:04-0400

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

TRQ\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 =TCQ\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 = QP(PQ)=2[3040]=1.5\frac{\partial Q}{\partial P} (\frac{P}{Q} ) = 2[\frac{30}{40}] = 1.5


Then plug 1.5 into the markup rule

P = MC[11E]=10[111.5]=10[13]=30\frac{MC}{[1 - \frac{1}{E}] } = \frac{10}{[1 - \frac{1}{1.5}] } = \frac{10}{[ \frac{1}{3}] } = 30


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