Final answer:
In an exchange economy, Person A starts with 128,000 units of x and zero units of y, while Person B starts with zero units of x and 16,000 units of y. To find the utility-maximizing quantities of goods x and y for each person, we need to compare the utilities of different combinations.
Step-by-step explanation:
In an exchange economy, Person A starts with 128,000 units of x and zero units of y, while Person B starts with zero units of x and 16,000 units of y. The utility functions are given as WA = X^A * Y^A and WB = X^B * Y^B. Here, X and Y represent the quantities of goods x and y, and A and B represent the exponents.
To find the utility-maximizing quantities of goods x and y for each person, we need to compare the utilities of different combinations. Since the utility functions rely on exponents, it would be helpful to express the quantities of goods as powers of 2 for easier comparison.
For Person A, 128,000 units of x can be written as 2^17, and 16,000 units of y can be written as 2^14. Substituting these values into the utility function, we get:
WA = (2^17)^A * (2^14)^A = 2^(17A) * 2^(14A) = 2^(31A).
Since B starts with zero units of x and 16,000 units of y, WB = (2^0)^B * (2^14)^B = 2^(0B) * 2^(14B) = 2^(14B).
Comparing the utilities, we have WA = 2^(31A) and WB = 2^(14B). To maximize utility, we need to find the values of A and B that will make the exponents as high as possible. However, to stay within the quantities of goods available, we need to ensure that the exponents are not greater than 17 for A and 14 for B.
Therefore, the utility-maximizing quantities are A = 14 and B = 17. This means that Person A should consume all 128,000 units of x and 16,000 units of y, while Person B should consume zero units of x and all 16,000 units of y.