Microeconomics, Module 11: “Monopoly” (Chapter 10)
Taxes, Price, and Quantity
(The attached PDF file has better formatting.)
Linear Demand Curves and Excise Taxes
A monopolist with constant marginal costs, MC = k, faces a linear demand curve Q = α – ßP, so ∂Q/∂P = –ß. We want to find the effect on the equilibrium price P* of a $1 per unit excise tax.
A. Rewrite the demand curve as P in terms of Q.
B. What is the total revenue curve? (Total revenue is P × Q.)
C. What is the marginal revenue curve? (Marginal revenue is the partial derivative of total revenue with respect to quantity.)
D. What is the monopoly quantity? (Equate marginal revenue and marginal cost.)
E. What is the monopoly price? (Find the price from the demand curve.)
F. What is the marginal cost with the excise tax?
G. What is the monopoly price with the excise tax?
H. What is the monopoly quantity with the excise tax?
Solution:
Part A: The market demand curve is Q = α – ßP, so ∂Q/∂P = –ß. We rewrite the demand curve as P = α/ß – Q/ß.
Part B: Total revenue = P × Q = αQ/ß – Q2/ß.
Part C: Marginal revenue = ∂(Total Revenue) / ∂Q = α/ß – 2Q/ß.
Part D: The monopolist produces a quantity where marginal revenue equals marginal cost. The marginal cost is a constant k, so
α/ß – 2Q/ß = k ➾ α/ß – k = 2Q/ß ➾ Q = ß/2 × (α/ß – k) = α/2 – kß/2.
Part E: The monopoly price is P = α/ß – Q/ß = α/ß – (α/2 – kß/2)/ß = α × (1/ß – 1/2) + k/2.
Part F: An excise tax of $1 increases the marginal cost by $1, so the monopoly price is
P = α × (1/ß – 1/2) + (k+1)/2.
An excise tax of $1 increases the price by $1/2.
Part G: An excise tax of $1 increases the marginal cost by $1, so
α/ß – 2Q*/ß = k+1 ➾ α/ß – (k+1) = 2Q/ß ➾ Q = ß/2 × (α/ß – (k+1)) = α/2 – (k+1)ß/2.
The decrease in quantity for an excise tax of $1 is ß/2.
● In a competitive market, an excise tax of $1 with a flat marginal cost curve increases the price by $1 and decreases quantity by β.
● In a monopoly market, an excise tax of $1 with a flat marginal cost curve increases the price by $0.50 and decreases quantity by β/2.