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 $\frac { - X } { 2 \sqrt { 500 - X ^ { 2 } } }$ .What is this slope at the utility-maximizing point?

Question

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The utility-maximizing point for perfect substitutes is where the marginal rate of substitution (MRS) equals the slope of the production possibilities frontier (PPF). For perfect substitutes, MRS = -1. Given the slope of the PPF is $\frac{-X}{2\sqrt{500-X^2}}$, at the utility-maximizing point, this slope must equal -1, matching choice B.

Suppose Goods X and Y Are Produced Along a Production −X2500−X2\frac { - X } { 2 \sqrt { 500 - X ^ { 2 } } }2500−X2​−X​

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