In: Economics
Suppose a monopolist faces the following demand curve:
Q=200-5P
also, the long run total cost of the monopolist is given by
TC=20+2Q-.5Q^2
a) what is the monopolist’s MC function?
b)what is the monopolist MR function?
c) what is the monopolist profit maximizing level of
output?
d) what is monopolist profit maximizing level of price?
e) how much profit is the monopoly firm is earning?
f) what is the value of consumer surplus under monopoly?
g) what is the value of the deadweight loss when the market is a
monopoly?
h) what is the value of the Lerner index?
Q = 200 – 5P (It must be P = 200 - 5Q). I am taking it as P = 200 - 5Q because when we draw MR curve of Q = 200 - 5P, it is never negative.
P = 200 - 5Q
Total Revenue = P * Q = 200Q - 5Q^2
TC = 20 + 2Q + 0.5Q2 (This must be a positive function because a negative sign of 0.5Q2 will make total cost negative after 8 units of production).
a) MC (first derivative of total cost with respect to Q) = 2 + Q
b) Marginal Revenue (First derivative of total revenue with respect to Q) = 200 - 10Q
c) Monopolist profit maximizing situation occurs when MR = MC
200 - 10Q = 2 + Q
198 = 11Q
Q = 18
d) At this quantity, Price monopolist charge is 36.4
e) Profit = [(Price - Average Total Cost) * Quantity sold]
Average total cost = Total Cost / Q
Average total cost = (20 / Q) + 2 - 0.5Q
Average total cost at Q = 18 is -5.88
Profit = (36.4 + 5.88) * 18 = 761.04
e) Consumer surplus is the area of portion A + B whose sum is (1/2) * (200 - 36.4) * (18 - 0) = 1,472.4
f) Deadweight loss is the area of triangle C whose sum is (1/2) * (33 - 18) * (36.4 - 20) = 123
g) Learner Index = [(P - MC) / P] = [(36.4 - 20) / 36.4] = 0.45