Suppose that on a fixed budget of $400 you buy x units of product A purchased at $20 apiece and y units of product B purchased at $20 apiece. For the utility function find x and y that maximize the utility function.

Question

Suppose that on a fixed budget of $400 you buy x units of product A purchased at $20 apiece and y units of product B purchased at $20 apiece. For the utility function  find x and y that maximize the utility function.

Quizplus · Accepted Answer

To maximize the utility function, we need to maximize the expression $x^{0.5}y^{0.5}$. Using the budget constraint, we can rewrite it as follows:
$$400 = 20x + 20y \Rightarrow y = 20 - x$$
Substituting this into our expression, we get:
$$U(x) = x^{0.5}(20-x)^{0.5}$$
Now, we can use calculus to find the critical points:
$$U'(x) = \frac{10-x}{\sqrt{x(20-x)}}$$
Setting this equal to zero and solving for x, we get $x = 10$, which is a maximum since $U''(x) < 0$ for $0 < x < 20$. Substituting this back into $y = 20 - x$, we get $y = 10$.
Therefore, the values that maximize the utility function are $x= 10$ and $y=10$, which correspond to choice B.

Suppose That on a Fixed Budget of $400 You Buy