Microeconomics Mod 4 elasticity practice exam questions

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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