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?

P2.6SOLUTION

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 Dogs could use the estimated linear market demand curve to estimate the quantity demanded during the same marketing period for ticket prices in the range from $5 to $15 per ticket, using $1 increments:

|Price |Quantity |TR=P×Q |

|$5 | | 30,000 |

| |6,000 | |

|6 | | 33,600 |

| |5,600 | |

|7 | | 36,400 |

| |5,200 | |

|8 | | 38,400 |

| |4,800 | |

|9 | | 39,600 |

| |4,400 | |

|10 | | 40,000 |

| |4,000 | |

|11 | | 39,600 |

| |3,600 | |

|12 | | 38,400 |

| |3,200 | |

|13 | | 36,400 |

| |2,800 | |

|14 | | 33,600 |

| |2,400 | |

|15 | | 30,000 |

| |2,000 | |

From the table, the revenue-maximizing ticket price is $10. This is also the profit-maximizing ticket price if variable costs and, hence, marginal costs are negligible. The pricing promotion resulted in declining revenues, and the $7 price results in an activity level that is above the revenue-maximizing output. Because the marginal cost of fan attendance cannot be less than zero, the profit-maximizing price cannot be less than the revenue-maximizing price of $10.

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