By NEAS - 11/27/2022 2:48:04 PM
Microeconomics Mod 4 elasticity practice exam questions
(The attached PDF file has better formatting.)
A monopolist faces a linear demand curve, produces the monopoly quantity, and sets monopoly prices.
● At price P = 4.58, total revenue is 254.469
● At price P = 8.86, total revenue is 319.731
The demand curve is linear: Q = α – β × P. By convention, this posting uses a positive value for β with the negative sign shown separately.
Question 4.1: Linear demand curve
What are the values of the α and β parameters of the linear demand curve?
Answer 4.1: Total revenue is price × quantity, giving
● 4.58 × (α – β × 4.58) = 254.469
● 8.86 × (α – β × 8.86) = 319.731
Solving the simultaneous linear equations gives
8.86 × 4.58 × (α – β × 4.58) = 8.86 × 254.469
4.58 × 8.86 × (α – β × 8.86) = 4.58 × 319.731
using the values
8.86 × 4.58 = 40.579
8.86 × 8.86 × 4.58 = 359.528
8.86 × 4.58 × 4.58 = 185.851
40.579 α – 185.851 β = 8.86 × 254.469 = 2,254.595
40.579 α – 359.528 β = 4.58 × 319.731 = 1,464.368
β = (2,254.595 – 1,464.368) / (359.528 – 185.851) = 4.550
α = (185.851 β + 2,254.595) / 40.579 = (185.851 × 4.550 + 2,254.595) / 40.579 = 76.400
The demand curve is Q = 76.400 – 4.550 × P.
Question 4.2: Price, quantity, and total revenue
What is the quantity at price P = 12.82?
Answer 4.2: We derive the quantity from the demand curve:
Q = 76.400 – 4.550 × 12.82 = 18.069
Question 4.3: Price elasticity of demand
What is the price elasticity of demand at price P = 12.82?
Answer 4.3: η = the price elasticity of demand = ∂Q / ∂P × P/Q = –β × P / Q = -4.550 × 12.82 / 18.069 = -3.228
Question 4.4: Marginal revenue and price
What is the marginal revenue at price P = 12.82?
Answer 4.4: Use the relation MR = P × (1 – 1 / |η| ) to give 12.82 × (1 – 1/3.228) = 8.849
Question 4.5: Marginal revenue and quantity
At what quantity is the marginal revenue equal to zero?
Answer 4.5: For a linear demand curve, the marginal revenue curve is twice as steep as the demand curve, so marginal revenue equals zero half-way between Q = 0 and Q = α.
½ × α = ½ × 76.400 = 38.200
Question 4.6: Price and elasticity
At what price is the price elasticity of demand equal to -1.30?
Answer 4.6: Solve for price as a function of the elasticity of demand and the parameters of the demand curve:
η = the price elasticity of demand = ∂Q / ∂P × P/Q = –β × P / (α – β × P).
1/η = –(α – β × P) / (β × P) = 1 – α / (β × P)
1 – 1/η = α / (β × P)
P = (α / β ) × η / (η – 1) = (76.400 / 4.550) × -1.30 / (-1.30 – 1) = (76.400 / 4.550) × 1.30 / (1 + 1.30) = 9.491
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By NEAS - 12/9/2024 9:31:06 PM
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