Question

Price cap monopoly question

Imagine a rm called Bapple that is the monopoly in the market for smartwatches, with
cost-function C(Q) = 99Q2 +20000. Imagine the inverse demand function for smartwatches
is p(Q) = 2000?Q. The government has decided it would ensure that there is no deadweight
loss in this market for smartwatches by setting a price cap on Bapple.

A. At what price should the government cap smartwatch sales?
B.What are the new post-price cap equilibrium price and equilibrium quantity?
C. What is Bapple's new pro t at the equilibrium?

D. Prove that this new pro t level is a global maximum.
E. Show the new equilibrium price and equilibrium quantity graphically. Include
the original and regulated inverse demand curves, firm's marginal revenue curve,
and firm's marginal cost curve.

F. What are consumer surplus, producer surplus, and deadweight loss at the
equilibrium? How have these quantities changed from the no-tax case in the
monopoly question?

F. What are consumer surplus, producer surplus, and deadweight loss at the
equilibrium? How have these quantities changed from the no-tax case in the
monopoly question?

Answer 1

A).

Consider the given problem here the demand and the cost functions are given by.

=> P = 2000 – q, and “C = 20,000 + 99*q^2”, => MC = 99*2*q = 198*q”, => MC = 198*q.

So, at “P=MC” the output will be efficient and the dead weight loss is will be “0”.

=> P = MC, => 2000 - q = 198*q, => q = 2,000/199 = 10.05. So, at “q=10.05” the corresponding “P” is given by, “P= 2000 - q = 1,989.9”.

So, here the government should cap the price at “P=$1,989.9”.

B).

The “post-cap” pricing and the quantity demanded are given by, “P = $1,989.5” and “q = 10.05”.

C).

The profit of the monopolist is given by.

=> A = P*q - C = 1,989.9*10.05 - (20,000 + 99*10.05^2) = $19,998.495 - ($29,999.25) = (-$10,000.75) < 0. So, here the monopolist is incurring loss at this new cap price.

D).

So, here the profit function is given by, “A= TR – TC", => FOC for maximization require.

=> dA/dq = 0, => d(TR)/dq - d(TC)/dq = 0, => MR – MC = 0, MR = MC. Now, the SOC require “d^2A/dq^2 < 0, => d(MR)/dq – d(MC)/dq < 0, => d(MR)/dq < d(MC)/dq, => “MC” must cut the MR from below.

Now, in this case as the government cap the price, => the new MR is “P=MR=1,989.9”, => MC cut the “MR” from below at “q=10.05”, => the intersection point is the globally maximum, => given the MR the profit is maximum.