49. Marginal Productivity A manufacturer estimates that produc- tion (in hundreds of units) is a function of the amounts $$ x $$ and $$ y $$ of labor and capital used, as follows. $$ f(x, y)=\left(\frac{1}{4} x^{-1 / 4}+\frac{3}{4} y^{-1 / 4}\right)^{-4} $$ a. Find the number of units produced when 16 units of labor and 81 units of capital are utilized. b. Find and interpret $$ f_{x}(16,81) $$ and $$ f_{y}(16,81) $$. c. What would be the approximate effect on production of increasing labor by 1 unit from 16 units of labor with 81 units of capital? 50. Marginal Productivity The production function $$ z $$ for the United States was once estimated as where $$ x $$ stands for the amount of labor and $$ y $$ the amount of capital. Find the marginal productivity of labor and of capital.

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Greene Mccarthy · Accepted Answer

For problem 49, part a: The number of units produced when 16 units of labor and 81 units of capital are utilized is approximately 5067 units. For part b: The partial derivatives $$ f_x(16, 81) $$ and $$ f_y(16, 81) $$ represent the marginal productivities of labor and capital at those specific inputs. For part c: The approximate effect on production of increasing labor by 1 unit is given by $$ f_x(16, 81) $$. For problem 50, the marginal productivity of labor and capital can be calculated once the specific production function $$ z $$ is provided.

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