Answer to Question #256086 in Calculus for Anuj

Question

A cellular phone company has the following production function for a smart phone: p(x, y) =

50x

2

3 y

1

3 where p is the number of units produced with x units of labor and y units of capital. a)

Find the number of units produced with 125 units of labor and 64 units of capital. b) Find the

marginal productivities (Hints: Partial derivatives). c) Evaluate the marginal productivities at

x = 125 and y = 64.

Answer 1

Solution:

Let the production function be $p(x,y)=50x^{2/3}y^{1/3}$

(a)

$p(125,64)=50(125)^{2/3}(64)^{1/3} \\ =50(25)(4) \\=5000\ units$

(b)

Marginal productivity of labor$=\dfrac{\partial p}{\partial x}=p_x$

$=50\cdot\dfrac23x^{-1/3}y^{1/3}=\dfrac{100y^{1/3}}{3x^{1/3}}$

Marginal productivity of capital$=\dfrac{\partial p}{\partial y}=p_y$

$=50\cdot\dfrac13x^{2/3}y^{-2/3}=\dfrac{50x^{2/3}}{3y^{2/3}}$

(c)

Marginal productivity of labor$=p_x(125,64)$

$=\dfrac{100(64)^{1/3}}{3(125)^{1/3}} \\=\dfrac{100\times 4}{3\times 5} \\=26 \dfrac23$

Marginal productivity of capital$=p_y(125,64)$

$=\dfrac{50(125)^{2/3}}{3(64)^{2/3}} \\=\dfrac{50\times25}{3\times16} \\=26 \dfrac1{24}$