To find the town's total demand function, we'll need to combine the individual demand functions for the two groups: college students and other town residents.
First, let's define the inverse demand functions for each group:
- For college students: [tex]\( p = 120 - Q_1 \)[/tex]
- For other town residents: [tex]\( p = 120 - 2Q_2 \)[/tex]
Next, we need to express the quantity demanded ([tex]\(Q_1\)[/tex] and [tex]\(Q_2\)[/tex]) in terms of the price ([tex]\(p\)[/tex]) for each group by solving these equations:
1. For college students:
[tex]\[ p = 120 - Q_1 \][/tex]
Rearrange to solve for [tex]\(Q_1\)[/tex]:
[tex]\[ Q_1 = 120 - p \][/tex]
2. For other town residents:
[tex]\[ p = 120 - 2Q_2 \][/tex]
Rearrange to solve for [tex]\(Q_2\)[/tex]:
[tex]\[ 2Q_2 = 120 - p \][/tex]
[tex]\[ Q_2 = \frac{120 - p}{2} \][/tex]
Now, the total demand [tex]\(Q_{total}\)[/tex] is the sum of [tex]\(Q_1\)[/tex] and [tex]\(Q_2\)[/tex]:
[tex]\[ Q_{total} = Q_1 + Q_2 \][/tex]
Substitute the expressions we found for [tex]\(Q_1\)[/tex] and [tex]\(Q_2\)[/tex]:
[tex]\[ Q_{total} = (120 - p) + \left( \frac{120 - p}{2} \right) \][/tex]
Combine the terms:
[tex]\[ Q_{total} = 120 - p + \frac{120 - p}{2} \][/tex]
[tex]\[ Q_{total} = 120 - p + 60 - \frac{p}{2} \][/tex]
Simplify the expression:
[tex]\[ Q_{total} = 180 - p - \frac{p}{2} \][/tex]
[tex]\[ Q_{total} = 180 - \frac{3p}{2} \][/tex]
Therefore, the town's total demand function is:
[tex]\[ Q_{total} = 180 - \frac{3p}{2} \][/tex]
