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Sagot :
To determine how Beth should allocate her [tex]$\$[/tex]25[tex]$ budget to maximize her utility, we must consider both nail polish (with a fixed price of $[/tex]\[tex]$5$[/tex] per bottle) and lipstick. The price of lipstick can be either [tex]$\$[/tex]10[tex]$ or $[/tex]\[tex]$5$[/tex]. We need to look at the marginal utility (MU) and the marginal utility per dollar (MU/P) for both goods.
Here is the provided data:
### Table A: Nail Polish
[tex]\[ \begin{array}{ccc} \hline & \text{Nail Polish} & MU & \frac{MU}{P} \\ \hline 0 & - & - \\ 1 & 45 & 9 \\ 2 & 40 & 8 \\ 3 & 30 & 6 \\ 4 & 15 & 3 \\ 5 & 5 & 1 \\ \hline \end{array} \][/tex]
### Table B: Lipstick - Two Scenarios
1. When Price of Lipstick ([tex]$P_L$[/tex]) = [tex]$\$[/tex]10[tex]$: \[ \begin{array}{ccc} \hline & \text{Lipstick} & MU & \frac{MU}{P_L} \\ \hline 0 & - & - \\ 1 & 40 & 4 \\ 2 & 30 & 3 \\ 3 & 20 & 2 \\ 4 & 10 & 1 \\ 5 & 5 & 0.5 \\ \hline \end{array} \] 2. When Price of Lipstick ($[/tex]P_L[tex]$) = $[/tex]\[tex]$5$[/tex]:
[tex]\[ \begin{array}{ccc} \hline & \text{Lipstick} & MU & \frac{MU}{P_L} \\ \hline 0 & - & - \\ 1 & 50 & 10 \\ 2 & 40 & 8 \\ 3 & 30 & 6 \\ 4 & 20 & 4 \\ 5 & 10 & 2 \\ \hline \end{array} \][/tex]
### Analysis:
#### Scenario 1: Lipstick Price = [tex]$10 - Combining Marginal Utilities per Dollar (Nail Polish + Lipstick at $[/tex]10):
- [tex]\( \frac{MU}{P} \)[/tex] for Nail Polish: 9, 8, 6, 3, 1
- [tex]\( \frac{MU}{P} \)[/tex] for Lipstick: 4, 3, 2, 1, 0.5
- Combined Marginal Utility per Dollar: 9, 8, 6, 3, 1, 4, 3, 2, 1, 0.5
- Sorted Marginal Utility per Dollar (Descending Order):
- 9, 8, 6, 4, 3, 3, 2, 1, 1, 0.5
- Maximizing Utility Within [tex]$\$[/tex]25[tex]$ Budget: - Begin with the highest $[/tex]\frac{MU}{P}[tex]$ and buy items until the budget is exhausted: 1. Buy 1st Nail Polish: MU = 45, Cost = $[/tex]5, Remaining Budget = [tex]$20 2. Buy 2nd Nail Polish: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$15 3. Buy 3rd Nail Polish: MU = 30, Cost = $[/tex]5, Remaining Budget = [tex]$10 4. Buy 1st Lipstick: MU = 40, Cost = $[/tex]10, Remaining Budget = [tex]$0 - Total Utility: 45 + 40 + 30 + 40 = 155 utils - Items Bought: 3 Nail Polishes, 1 Lipstick - Budget Left: $[/tex]0
#### Scenario 2: Lipstick Price = [tex]$5 - Combining Marginal Utilities per Dollar (Nail Polish + Lipstick at $[/tex]5):
- [tex]\( \frac{MU}{P} \)[/tex] for Nail Polish: 9, 8, 6, 3, 1
- [tex]\( \frac{MU}{P} \)[/tex] for Lipstick: 10, 8, 6, 4, 2
- Combined Marginal Utility per Dollar: 9, 8, 6, 3, 1, 10, 8, 6, 4, 2
- Sorted Marginal Utility per Dollar (Descending Order):
- 10, 9, 8, 8, 6, 6, 4, 3, 2, 1
- Maximizing Utility Within [tex]$\$[/tex]25[tex]$ Budget: - Begin with the highest $[/tex]\frac{MU}{P}[tex]$ and buy items until the budget is exhausted: 1. Buy 1st Lipstick: MU = 50, Cost = $[/tex]5, Remaining Budget = [tex]$20 2. Buy 1st Nail Polish: MU = 45, Cost = $[/tex]5, Remaining Budget = [tex]$15 3. Buy 2nd Nail Polish: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$10 4. Buy 2nd Lipstick: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$5 5. Buy 3rd Lipstick: MU = 30, Cost = $[/tex]5, Remaining Budget = [tex]$0 - Total Utility: 50 + 45 + 40 + 40 + 30 = 205 utils - Items Bought: 2 Nail Polishes, 3 Lipsticks - Budget Left: $[/tex]0
### Conclusion:
- When the price of lipstick is [tex]$\$[/tex]10[tex]$, Beth's maximum attainable utility is 155 utils by buying 3 bottles of nail polish and 1 tube of lipstick, with no budget remaining. - When the price of lipstick is $[/tex]\[tex]$5$[/tex], Beth's maximum attainable utility is 205 utils by buying 2 bottles of nail polish and 3 tubes of lipstick, with no budget remaining.
Here is the provided data:
### Table A: Nail Polish
[tex]\[ \begin{array}{ccc} \hline & \text{Nail Polish} & MU & \frac{MU}{P} \\ \hline 0 & - & - \\ 1 & 45 & 9 \\ 2 & 40 & 8 \\ 3 & 30 & 6 \\ 4 & 15 & 3 \\ 5 & 5 & 1 \\ \hline \end{array} \][/tex]
### Table B: Lipstick - Two Scenarios
1. When Price of Lipstick ([tex]$P_L$[/tex]) = [tex]$\$[/tex]10[tex]$: \[ \begin{array}{ccc} \hline & \text{Lipstick} & MU & \frac{MU}{P_L} \\ \hline 0 & - & - \\ 1 & 40 & 4 \\ 2 & 30 & 3 \\ 3 & 20 & 2 \\ 4 & 10 & 1 \\ 5 & 5 & 0.5 \\ \hline \end{array} \] 2. When Price of Lipstick ($[/tex]P_L[tex]$) = $[/tex]\[tex]$5$[/tex]:
[tex]\[ \begin{array}{ccc} \hline & \text{Lipstick} & MU & \frac{MU}{P_L} \\ \hline 0 & - & - \\ 1 & 50 & 10 \\ 2 & 40 & 8 \\ 3 & 30 & 6 \\ 4 & 20 & 4 \\ 5 & 10 & 2 \\ \hline \end{array} \][/tex]
### Analysis:
#### Scenario 1: Lipstick Price = [tex]$10 - Combining Marginal Utilities per Dollar (Nail Polish + Lipstick at $[/tex]10):
- [tex]\( \frac{MU}{P} \)[/tex] for Nail Polish: 9, 8, 6, 3, 1
- [tex]\( \frac{MU}{P} \)[/tex] for Lipstick: 4, 3, 2, 1, 0.5
- Combined Marginal Utility per Dollar: 9, 8, 6, 3, 1, 4, 3, 2, 1, 0.5
- Sorted Marginal Utility per Dollar (Descending Order):
- 9, 8, 6, 4, 3, 3, 2, 1, 1, 0.5
- Maximizing Utility Within [tex]$\$[/tex]25[tex]$ Budget: - Begin with the highest $[/tex]\frac{MU}{P}[tex]$ and buy items until the budget is exhausted: 1. Buy 1st Nail Polish: MU = 45, Cost = $[/tex]5, Remaining Budget = [tex]$20 2. Buy 2nd Nail Polish: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$15 3. Buy 3rd Nail Polish: MU = 30, Cost = $[/tex]5, Remaining Budget = [tex]$10 4. Buy 1st Lipstick: MU = 40, Cost = $[/tex]10, Remaining Budget = [tex]$0 - Total Utility: 45 + 40 + 30 + 40 = 155 utils - Items Bought: 3 Nail Polishes, 1 Lipstick - Budget Left: $[/tex]0
#### Scenario 2: Lipstick Price = [tex]$5 - Combining Marginal Utilities per Dollar (Nail Polish + Lipstick at $[/tex]5):
- [tex]\( \frac{MU}{P} \)[/tex] for Nail Polish: 9, 8, 6, 3, 1
- [tex]\( \frac{MU}{P} \)[/tex] for Lipstick: 10, 8, 6, 4, 2
- Combined Marginal Utility per Dollar: 9, 8, 6, 3, 1, 10, 8, 6, 4, 2
- Sorted Marginal Utility per Dollar (Descending Order):
- 10, 9, 8, 8, 6, 6, 4, 3, 2, 1
- Maximizing Utility Within [tex]$\$[/tex]25[tex]$ Budget: - Begin with the highest $[/tex]\frac{MU}{P}[tex]$ and buy items until the budget is exhausted: 1. Buy 1st Lipstick: MU = 50, Cost = $[/tex]5, Remaining Budget = [tex]$20 2. Buy 1st Nail Polish: MU = 45, Cost = $[/tex]5, Remaining Budget = [tex]$15 3. Buy 2nd Nail Polish: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$10 4. Buy 2nd Lipstick: MU = 40, Cost = $[/tex]5, Remaining Budget = [tex]$5 5. Buy 3rd Lipstick: MU = 30, Cost = $[/tex]5, Remaining Budget = [tex]$0 - Total Utility: 50 + 45 + 40 + 40 + 30 = 205 utils - Items Bought: 2 Nail Polishes, 3 Lipsticks - Budget Left: $[/tex]0
### Conclusion:
- When the price of lipstick is [tex]$\$[/tex]10[tex]$, Beth's maximum attainable utility is 155 utils by buying 3 bottles of nail polish and 1 tube of lipstick, with no budget remaining. - When the price of lipstick is $[/tex]\[tex]$5$[/tex], Beth's maximum attainable utility is 205 utils by buying 2 bottles of nail polish and 3 tubes of lipstick, with no budget remaining.
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