Landsburg Mod 12 Ch10 third degree price discrimination practice exam questions
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A monopolist sells to two sets of consumers, Group Y and Group Z.
● The demand curve per person for Group Y is Q = 18.7 – 2.34 × P.
● The demand curve per person for Group Z is Q = 24.4 – 4.74 × P.
The quantity Q and the price P may be any positive real numbers, not just integers.
The marginal cost is MC = 0. Group Y has 158 persons and Group Z has 242 persons.
Question 12.1: Combined demand curve intercept
If the price is zero, what is the total quantity demanded? This is the α parameter of the combined demand curve.
Answer 12.1: The quantity demanded by Group Y is Q = 18.7 × 158 and the quantity demanded by Group Z is Q = 24.4 × 242, so the total quantity demanded is Q = 18.7 × 158 + 24.4 × 242 = 8,859.4
Question 12.2: Combined demand curve slope
If the price increases by one, what is the decrease in the total quantity demanded? This is the β parameter of the combined demand curve.
Answer 12.2: If the price increases by one unit, the quantity demanded by Group Y decreases by Q = 2.34 × 158 and the quantity demanded by Group Z decreases by Q = 4.74 × 242, so the total quantity demanded decreases by Q = 2.34 × 158 + 4.74 × 242 = 1,516.8.
Question 12.3: marginal revenue
If price discrimination is not allowed (all persons are charged the same price), what is the marginal revenue curve?
Answer 12.3: For a linear demand curve (Q = α – β × P), the marginal revenue curve has the same intercept as the demand curve and is twice as steep (twice the slope), or MR = α – 2β × P =
MR = 8,859.4 – 2 × 1,516.8 × P
Question 12.4: Equilibrium price
If price discrimination is not allowed (all persons are charged the same price), what is the price charged? (This is a monopoly, not a competitive market; assume the monopoly price is charged.)
Answer 12.4: The monopoly price is where marginal revenue equals marginal cost. For simplicity, marginal cost equals zero in this problem, so the monopoly price is where marginal revenue equals zero, or
MR = 8,859.4 – 2 × 1,516.8 × P = 0 ➾
P = 8,859.4 / (2 × 1,516.8) = 2.92
Question 12.5: Equilibrium quantity Group Y
If price discrimination is not allowed (all persons are charged the same price), what is the quantity bought by Group Y?
Answer 12.5: The price charged is 2.92, so the quantity bought by Group Y is
158 persons × (18.7 – 2.34 × P) per person = 158 × (18.7 – 2.34 × 2.92) = 1,875.02
Question 12.6: Equilibrium quantity Group Z
If price discrimination is not allowed (all persons are charged the same price), what is the quantity bought by Group Z?
Answer 12.6: The price charged is 2.92, so the quantity bought by Group Z is
242 persons × (24.4 – 4.74 × P) per person = 242 × (24.4 – 4.74 × 2.92) = 2,555.33
Question 12.7: Consumers’ surplus
If price discrimination is not allowed (all persons are charged the same price), what is consumers’ surplus for Groups Y and Z?
Answer 12.7: Consumers’ surplus is the area under the demand curve, down to the equilibrium price, and out to the equilibrium quantity. Since the demand curve is linear, consumers’ surplus is a right triangle with three vertices (quantity is on the horizontal axis; price is on the vertical axis):
The demand curve per person for Group Y is Q = 18.7 – 2.34 × P.
● When Q = 0, P = 18.7 / 2.34 = 7.99 (from the demand curve).
● The equilibrium price is 2.92.
● The equilibrium quantity 1,875.02 units for Group Y.
The vertices (Q,P) of the right triangle are (0, 7.99), (0, 2.92), (1,875.02, 2.92).
Consumers’ surplus is ½ × 1,875.02 × (7.99 – 2.92) = 4,753.18.
The demand curve per person for Group Z is Q = 24.4 – 4.74 × P.
● When Q = 0, P = 24.4 / 4.74 = 5.15 (from the demand curve).
● The equilibrium price is 2.92.
● The equilibrium quantity 2,555.33 units for Group Z.
The vertices (Q,P) of the right triangle are (0, 5.15), (0, 2.92), (2,555.33, 2.92).
Consumers’ surplus is ½ × 2,555.33 × (5.15 – 2.92) = 2,849.19
Question 12.8: Producers’ surplus
If price discrimination is not allowed (all persons are charged the same price), what is producers’ surplus for Groups Y and Z?
Answer 12.8: Producers’ surplus is the area above the short-run supply curve, up to the equilibrium price, and out to the equilibrium quantity. The short-run supply curve for a monopolist is the marginal cost curve, which is flat at zero in this problem, so producers’ surplus is a rectangle with
● height = The equilibrium price of 2.92
● base = The equilibrium quantity of 1,875.02 units
for an area of 2.92 × 1,875.02 = 5,475.06 = producers’ surplus for Group Y, and
● height = The equilibrium price of 2.92
● base = The equilibrium quantity of 2,555.33 units
for an area of 2.92 × 2,555.33 = 7,461.56 = producers’ surplus for Group Z
Question 12.9: Price discrimination price and quantity
If the monopolist sells at different prices to Groups Y and Z, what is the price charged and the quantity sold to each group?
Answer 12.9: The monopoly price is where marginal revenue equals marginal cost. For simplicity, marginal cost equals zero in this problem, so the monopoly price is where marginal revenue equals zero, or
Group Y: MR = 18.7 – 2 × 2.34 × P = 0 ➾
P = 18.7 / (2 × 2.34 ) = 4.00
Group Z: MR = 24.4 – 2 × 4.74 × P = 0 ➾
P = 24.4 / (2 × 4.74) = 2.57
The quantity bought by each Group is derived from its demand curve and its size:
Group Y: 158 persons × (18.7 – 2.34 × P) per person = 158 × (18.7 – 2.34 × 4.00) = 1,475.72
Group Z: 242 persons × (24.4 – 4.74 × P) per person = 242 × (24.4 – 4.74 × 2.57) = 2,956.80
Question 12.10: Price discrimination consumers’ surplus
If the monopolist sells at different prices, what is consumers’ surplus for Group Y?
Answer 12.10: Consumers’ surplus is the area under the demand curve, down to the equilibrium price, and out to the equilibrium quantity. Since the demand curve is linear, consumers’ surplus is a right triangle with three vertices (quantity is on the horizontal axis; price is on the vertical axis):
The demand curve per person for Group Y is Q = 18.7 – 2.34 × P.
● When Q = 0, P = 18.7 / 2.34 = 7.99 (from the demand curve).
● The equilibrium price is 4.00.
● The equilibrium quantity is 1,475.72 units for Group Y.
The vertices (Q,P) of the right triangle are (0, 7.99), (0, 4.00), (1,475.72, 4.00).
Consumers’ surplus is ½ × 1,475.72 × (7.99 – 4.00) = 2,944.06
Question 12.11: Price discrimination consumers’ surplus
If the monopolist sells at different prices, what is consumers’ surplus for Group Z?
Answer 12.11: Consumers’ surplus is the area under the demand curve, down to the equilibrium price, and out to the equilibrium quantity. Since the demand curve is linear, consumers’ surplus is a right triangle with three vertices (quantity is on the horizontal axis; price is on the vertical axis):
The demand curve per person for Group Z is Q = 24.4 – 4.74 × P.
● When Q = 0, P = 24.4 / 4.74 = 5.15 (from the demand curve).
● The equilibrium price is 2.57
● The equilibrium quantity 2,956.80 units for Group Z.
The vertices (Q,P) of the right triangle are (0, 5.15), (0, 2.57), (2,956.80, 2.57).
Consumers’ surplus is ½ × 2,956.80 × (5.15 – 2.57) = 3,814.27
Question 12.12: Price discrimination producers’ surplus
If the monopolist sells at different prices, what is producers’ surplus for Groups Y and Z?
Answer 12.12: Producers’ surplus is the area above the short-run supply curve, up to the equilibrium price, and out to the equilibrium quantity. The short-run supply curve for a monopolist is the marginal cost curve, which is flat at zero in this problem, so producers’ surplus is a rectangle with
● height = The equilibrium price of 4.00
● base = The equilibrium quantity of 1,475.72 units
for an area of 4.00 × 1,475.72 = 5,902.88 = producers’ surplus for Group Y, and
● height = The equilibrium price of 2.57
● base = The equilibrium quantity of 2,956.80 units
for an area of 2.57 × 2,956.80 = 7,598.98 = producers’ surplus for Group Z