Answer :
Answer:
Charlie consumes 100 books and 25 wines at pareto-optimal allocation, is the right answer.
Explanation:
According to the question, book is denoted with b whereas, wine is denoted with w.
The Utility function of Martha is- [tex]U\left ( b,w \right )=6b+24w[/tex]
The Utility function for Charlie is- [tex]U\left ( b,w \right )=bw[/tex]
[tex]P_{b}[/tex] = $1 : [tex]P_{w}[/tex] = $4
At pareto-optimality,
[tex]MRS_{bw}=\frac{P_{b}}{P_{w}}\\\frac{w}{b}=\frac{1}{4}\\4w=b\\[/tex]
The budget constraint,
[tex]1.b+4.w=m=1\left ( 20 \right )+4\left ( 45 \right )\\b+4w=20+180\\4w+4w=200\\w=\frac{200}{8}=25\\\therefore b=100[/tex]
Therefore, it may be said that Charlie must consume 100 books and 25 wine at pareto-optimal allocation.