Question

Catalina Films produces video shorts using digital editing equipment (K) and editors (L).

 The firm has the production function Q=10L^0.5K^{0 .5}where Q is the hours of edited footage. The wage is $20, and the rental rate of capital is $5. The firm wants to produce 1000 units of output at the lowest possible cost. What is the optimal combination of editing equipment(K) and editors(L) that will minimize cost for Catalina Films?

Answer 1

Solution:

Q = 10L^0.5K^0.5

Derive MRTS: = $\frac{MP_{L} }{MP_{K}}$

MP_L = $\frac{\partial Q} {\partial L}$= 5L^-0.5K^0.5

MP_K = $\frac{\partial Q} {\partial L}$ = 5L^0.5K^-0.5

MRTS = 5L^-0.5K^0.5 $\div$ 5L^0.5K^-0.5 = $\frac{K}{L}$

$\frac{K}{L} = \frac{w}{r}$

w = 20

r = 5

$\frac{K}{L} = \frac{20}{5}$

K = 4L

Substitute in the production function:

Q = 10L^0.5K^0.5

Q = 1,000

1,000 = 10L^0.54L^0.5

L = 25

K = 4L = 4 $\times$ 25 = 100

The optimal combination of editing equipment(K) and editors(L) that will minimize cost for Catalina Films is:

Editing equipment’s (K) = 100

Editors (L) = 25