Question

There are two people in the economy: Martha and Charlie. There are two goods in the economy: books and wine. Martha is endowed with 110 books and 20 wine bottles, while Charlie is endowed with 20 books and 45 wine bottles. Denote books with, b, and wine with, w. Martha has the utility function: U(b,w) = 6b+24w. Charlie has the utility function: U(b,w) = bw. At a pareto optimal allocation, the price of books is $1, while the price of wine is $4. How many books does Charlie consume at a pareto-optimal allocation in the exchange equilibrium?

Answer 1

Charlie consumes 100 books and 25 wines at pareto-optimal allocation, is the right answer.

Step-by-step explanation:

According to the question, book is denoted with b whereas, wine is denoted with w.

The Utility function of Martha is-
`U\left ( b,w \right )=6b+24w`

The Utility function for Charlie is-
`U\left ( b,w \right )=bw`

`P_(b)` = $1 :
`P_(w)` = $4

At pareto-optimality,

`MRS_(bw)=(P_(b))/(P_(w))\\(w)/(b)=(1)/(4)\\4w=b\\`

The budget constraint,

`1.b+4.w=m=1\left ( 20 \right )+4\left ( 45 \right )\\b+4w=20+180\\4w+4w=200\\w=(200)/(8)=25\\\therefore b=100`

Therefore, it may be said that Charlie must consume 100 books and 25 wine at pareto-optimal allocation.