Managerial Economics Chapter 2 Hirschey
P2.6 Price and Total Revenue. The Portland Sea Dogs, the AA affiliate of the Boston Red Sox major league baseball team, have enjoyed a surge in popularity. During a recent home stand, suppose the club offered $5 off the $12 regular price of reserved seats, and sales spurted from 3,200 to 5,200 tickets per game.
A. Derive the function that describes the price/output relation with price expressed as a function of quantity (tickets sold). Also express tickets sold as a function of price.
B. Use the information derived in part A to calculate total revenues at prices in $1 increments from $5 to $15 per ticket. What is the revenue-maximizing ticket price? If variable costs are negligible, is this amount also the profit-maximizing ticket price?
P2.6 SOLUTION
A. When a linear demand curve is written as:
P = a + bQ
a is the intercept and b is the slope coefficient. Because 3,200 seats were sold at a regular price of $12 per game, and 5,200 seats were sold at the discount price of $7, two points on the firm’s linear demand curve are identified. Given this information, it is possible to identify the linear demand curve by solving the system of two equations with two unknowns, a and b:
12 = a + b(3,200) minus 7 = a + b(5,200) 5 = -2,000 b
b = -0.0025
By substitution, if b = -0.0025, then:
12 = a + b(3,200)
12 = a - 0.0025(3,200)
12 = a - 8
a = 20
With price expressed as a function of quantity, the reserved seat demand curve can be written:
P = $20 - $0.0025Q
Similarly, the number of tickets sold (quantity) can be expressed as a function of price:
P = $20 - $0.0025Q
0.0025Q = $20 - P
Q = 8,000 – 400P
This simple linear characterization of the firm’s demand curve can be used to profitably guide production, pricing and promotion decisions.
B. The Portland Sea
A. Derive the function that describes the price/output relation with price expressed as a function of quantity (tickets sold). Also express tickets sold as a function of price.
B. Use the information derived in part A to calculate total revenues at prices in $1 increments from $5 to $15 per ticket. What is the revenue-maximizing ticket price? If variable costs are negligible, is this amount also the profit-maximizing ticket price?
P2.6 SOLUTION
A. When a linear demand curve is written as:
P = a + bQ
a is the intercept and b is the slope coefficient. Because 3,200 seats were sold at a regular price of $12 per game, and 5,200 seats were sold at the discount price of $7, two points on the firm’s linear demand curve are identified. Given this information, it is possible to identify the linear demand curve by solving the system of two equations with two unknowns, a and b:
12 = a + b(3,200) minus 7 = a + b(5,200) 5 = -2,000 b
b = -0.0025
By substitution, if b = -0.0025, then:
12 = a + b(3,200)
12 = a - 0.0025(3,200)
12 = a - 8
a = 20
With price expressed as a function of quantity, the reserved seat demand curve can be written:
P = $20 - $0.0025Q
Similarly, the number of tickets sold (quantity) can be expressed as a function of price:
P = $20 - $0.0025Q
0.0025Q = $20 - P
Q = 8,000 – 400P
This simple linear characterization of the firm’s demand curve can be used to profitably guide production, pricing and promotion decisions.
B. The Portland Sea