The productivity of a certain country is P(x, y) = 1600 $x ^ { 1 / 4 }$ $y ^ { 3 / 4 }$ units, where x and y are the amounts of labor and capital utilized. What is the marginal productivity of capital when x = 16 and y = 625?
A) 1600
B) 5000
C) 480
D) 2500
E) none of these

Question

Quizplus · Accepted Answer

The marginal productivity of capital is found by taking the partial derivative of the productivity function P(x, y) with respect to y, and then substituting the given values of x and y. The partial derivative of P with respect to y is $\frac{\partial P}{\partial y} = 1600x^{1/4} \cdot \frac{3}{4}y^{-1/4}$. Substituting x = 16 and y = 625 gives $\frac{\partial P}{\partial y} = 1600(16)^{1/4} \cdot \frac{3}{4}(625)^{-1/4} = 1600 \cdot 2 \cdot \frac{3}{4} \cdot \frac{1}{5} = 480$.

The Productivity of a Certain Country Is P(x, Y) = x1/4x ^ { 1 / 4 }x1/4

The Productivity of a Certain Country Is P(x, Y) = $x ^ { 1 / 4 }$