Suppose that the inverse demand function for movies is P = 120 – Q1 for college students and P = 120 – 2Q2 for other town residents. What is the town’s total demand function ( Q = Q1 + Q2 as a functiof P )?
The correct answer and explanation is :
The inverse demand functions for movies are given as:
- College students: ( P = 120 – Q_1 ) → Solving for ( Q_1 ):
[
Q_1 = 120 – P
] - Town residents: ( P = 120 – 2Q_2 ) → Solving for ( Q_2 ):
[
Q_2 = \frac{120 – P}{2}
]
Total demand function is the sum of both quantities:
[
Q = Q_1 + Q_2 = (120 – P) + \frac{120 – P}{2}
]
Finding a common denominator:
[
Q = \frac{2(120 – P)}{2} + \frac{120 – P}{2}
]
[
Q = \frac{2(120 – P) + (120 – P)}{2}
]
[
Q = \frac{3(120 – P)}{2}
]
[
Q = \frac{360 – 3P}{2}
]
Thus, the total demand function is:
[
Q = 180 – \frac{3}{2} P
]
Explanation:
The inverse demand functions describe the price consumers are willing to pay for a given quantity. The first function applies to college students, and the second applies to other town residents, where town residents are twice as responsive to price changes compared to students (since the coefficient of ( Q_2 ) is 2).
To determine the total demand, we express both demand quantities as functions of price and sum them up. Since town residents have a more elastic demand (higher sensitivity to price), their contribution is weighted accordingly when combined with the college students’ demand.
The resulting total demand equation combines both groups’ demand, showing how total quantity demanded changes with price. The slope of -( \frac{3}{2} ) indicates that for every $1 increase in price, total demand decreases by ( 1.5 ) units. This reflects the differing price sensitivities of the two consumer groups.