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

$$p(x,y)=50x^{2/3}y^{1/3}$$

a)

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

b)

$$\dfrac{\partial p}{\partial x}=\dfrac{100}{3}x^{-1/3}y^{1/3}$$

$$\dfrac{\partial p}{\partial y}=\dfrac{50}{3}x^{2/3}y^{-2/3}$$

c)

$$\dfrac{\partial p}{\partial x}(125,64)=\dfrac{100}{3}(125)^{-1/3}(64)^{1/3}=\dfrac{80}{3}$$

$$\dfrac{\partial p}{\partial y}(125,64)=\dfrac{50}{3}(125)^{2/3}(64)^{-2/3}=\dfrac{625}{24}$$