Suppose goods X and Y are produced along a production possibilities frontier 4X² + Y² = 500 and they are perfect substitutes such that U = X + Y.The slope of the production possibilities frontier is $\frac { - 4 X } { \sqrt { 500 - 4 X ^ { 2 } } }$ .What is this slope at the utility-maximizing point? A)0 B)-1 C)-4 D)-5

Question

Quizplus · Accepted Answer

To find the slope at the utility-maximizing point, we first need to find the marginal rate of substitution (MRS) of X for Y at that point. Since U = X + Y, we can rewrite the production possibilities frontier as simply X + Y = 500. Solving for Y, we get Y = 500 - X. Substituting this into the utility function, we get U = X + (500 - X) = 500.

The MRS is the absolute value of the slope of the indifference curve at the utility-maximizing point. Since U is a linear function of X and Y, the indifference curve is a straight line with slope of -1. Therefore, the MRS is always 1.

Now, we can use the fact that the MRS equals the ratio of the marginal products to find the slope of the production possibilities frontier at the utility-maximizing point. Since X and Y are perfect substitutes, the marginal products are equal. From the given information, the marginal product of X is the slope of the production possibilities frontier, which is $\frac{-4X}{\sqrt{500-4X^2}}$. Therefore, we have:

MRS = $\frac{\text{Marginal Product of X}}{\text{Marginal Product of Y}}$ = $\frac{-4X/\sqrt{500-4X^2}}{1}$ = 1

Solving for X, we get X = 125, and therefore Y = 375. So, the slope of the production possibilities frontier at the utility-maximizing point is:

$\frac{-4X}{\sqrt{500-4X^2}}$ = $\frac{-4(125)}{\sqrt{500-4(125)^2}}$ = -1

Therefore, the answer is (B).

Suppose Goods X and Y Are Produced Along a Production −4X500−4X2\frac { - 4 X } { \sqrt { 500 - 4 X ^ { 2 } } }500−4X2​−4X​

Suppose Goods X and Y Are Produced Along a Production $\frac { - 4 X } { \sqrt { 500 - 4 X ^ { 2 } } }$