In: Economics
Ken runs a barber shop. Given the popularity and location of the restaurant, he has a monopoly position in the market. The inverse market demand curve is given by Q = 120 – 2P. Ken has a total cost of TC = Q2. If he charges the same price to all customers, what are Ken’s profit-maximising price PM and quantity QM?
Profit-maximizing stage:
This is the stage where (MR = MC).
Let QM is Q and PM is P.
Given,
Q = 120 – 2P
By rearranging,
2P = 120 – Q
P = 120/2 – Q/2
P = 60 – 0.5Q ……………….. (Price function)
Hence TR is the product of P and Q.
TR = P × Q = (60 – 0.5Q) × Q
= 60Q – 0.5Q^2
Now MR is the derivative of TR with respect to Q.
TR = 60Q – 0.5Q^2
MR = (d/dQ) (60Q – 0.5Q^2)
= 60 – (0.5 × 2)Q^(2 – 1)
= 60 – 1Q
= 60 – Q
Again given,
TC = Q^2
MC is to be calculated by derivative of TC.
MC = (d/dQ) Q^2
= 2Q
Therefore, there would be the equality,
MR = MC
60 – Q = 2Q
60 = 2Q + Q
60 = 3Q
60/3 = Q
Q = 20
This is to be placed in the price function,
P = 60 – 0.5Q
= 60 – 0.5 × 20
= 60 – 10
= 50
Answers:
PM = 50
QM = 20