To determine how to best split the [tex]$20,000 between Jane and Fred in order to maximize their combined utility, we need to look at the table of potential allocations and their corresponding marginal utilities for the last dollar spent by each individual.
The table is given as:
\[
\begin{array}{|c|c|c|c|}
\hline
\text{Amount to Jane} & \text{Amount to Fred} & \text{MU of Jane's Last Dollar Spent} & \text{MU of Fred's Last Dollar Spent} \\
\hline
\$[/tex]5,000 & \[tex]$15,000 & 80 & 65 \\
\hline
\$[/tex]7,000 & \[tex]$13,000 & 70 & 70 \\
\hline
\$[/tex]9,000 & \[tex]$11,000 & 60 & 75 \\
\hline
\$[/tex]11,000 & \[tex]$9,000 & 50 & 80 \\
\hline
\$[/tex]13,000 & \[tex]$7,000 & 40 & 85 \\
\hline
\end{array}
\]
We need to calculate the combined marginal utility for each distribution and then compare them to find the maximum combined marginal utility.
Let's list the combined utilities:
1. For $[/tex](\[tex]$5,000, \$[/tex]15,000)[tex]$: $[/tex]80 + 65 = 145[tex]$
2. For $[/tex](\[tex]$7,000, \$[/tex]13,000)[tex]$: $[/tex]70 + 70 = 140[tex]$
3. For $[/tex](\[tex]$9,000, \$[/tex]11,000)[tex]$: $[/tex]60 + 75 = 135[tex]$
4. For $[/tex](\[tex]$11,000, \$[/tex]9,000)[tex]$: $[/tex]50 + 80 = 130[tex]$
5. For $[/tex](\[tex]$13,000, \$[/tex]7,000)[tex]$: $[/tex]40 + 85 = 125[tex]$
Comparing these combined utilities, the maximum combined utility is \(145\), which corresponds to allocating $[/tex]5,000 to Jane and [tex]$15,000 to Fred.
Therefore, the optimal allocation to maximize their combined utility is:
- Amount to Jane = $[/tex]5,000
- Amount to Fred = $15,000
