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Homework: Consumer Choice and Demand

4. Deriving Demand from a Marginal Utility Table with Two Goods

Beth has a [tex]$25 cosmetics budget that she uses to buy nail polish and lipstick. Assume the price of nail polish \([ P_N ]\) is fixed at $[/tex]5. Table A shows the marginal utility (MU) and marginal utility per dollar [tex]\(\left( \frac{MU}{P} \right)\)[/tex] she receives from the first through fifth bottles of nail polish she buys each month. Table B shows the same information for lipstick when the price of a tube of lipstick [tex]\([ P_L ]\)[/tex] is either [tex]$10 or $[/tex]5. Assume that Beth is a rational consumer who wants to maximize her utility.

\begin{tabular}{ccc}
\hline & Table A \\
Nail Polish & MU & [tex]\(\frac{MU}{P}\)[/tex] \\
(Bottles) & (Utils) & (If [tex]\(P = \$5\)[/tex]) \\
\hline 0 & - & - \\
1 & 45 & 9 \\
2 & 40 & 8 \\
3 & 30 & 6 \\
4 & 15 & 3 \\
5 & 5 & 1 \\
\end{tabular}

\begin{tabular}{ccc}
\hline & Table B \\
Lipstick & MU & [tex]\(\frac{MU}{P}\)[/tex] \\
(Tubes) & (Utils) & (If [tex]\(P = \$10\)[/tex]) & (If [tex]\(P = \$5\)[/tex]) \\
\hline 0 & - & - & - \\
1 & 50 & 5 & 10 \\
2 & 45 & 4.5 & 9 \\
3 & 35 & 3.5 & 7 \\
4 & 20 & 2 & 4 \\
5 & 10 & 1 & 2 \\
\end{tabular}



Answer :

GinnyAnswer
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.

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