A monopolist faces the inverse demand for its output:

Question

p = 30 – Q
The monopolist also has a constant marginal and average cost of $4/unit. The government is seeking ways to collect tax revenue from the monopolist and faces two proposals:
i. Impose a specific tax of t on the monopolist.
ii. Impose an ad valorem tax of a on the monopolist.
a. Suppose the government imposes a 20% ad valorem tax on the monopolist. What price and quantity does the monopolist choose and how much revenue does the government generate from the tax?
b. Rather than an ad valorem tax, what is the government's revenue from a specific tax of t imposed on the monopolist? Your answer should be in terms of t.
c. Show that a specific tax of $3.70/unit generates the same revenue as a 20% ad valorem tax (approximately).
d. Which tax has a greater distortion on the monopoly output?

Answer 1

a. With a 10% ad valorem tax, the government seeks to maximize:
Pi = (1 – 0.20 )(30 – Q)Q – 4Q
The first order condition is:
.80(30 – 2Q) – 4 = 0
Solving yields
Q* = 12.5
The price (from the demand curve) is:
p* = 17.5
The government's revenue from the tax is:
GR* = 43.75
b. A specific tax of t makes the monopolist's profit maximization:
Pi = (30 – Q)Q – (4 + t)Q
The first order condition is:
30 – 2Q – 4 – t = 0
Solving for quantity:
Q = (26 – t)/2 = 13 – t/2
The government's revenue is:
GR = (13 – t/2 )t
c. Find t* where:
(13 – t/2)t = 43.75.
Solving the quadratic:
t = 3.7
d. The output is only Q = 11.15 with the specific tax versus 12.5 with the ad valorem tax.