Question

A businessman uses K and L to produce X. Production function is: Q = 2K ( L – 2 )

PK = 600, PL = 300, TC = 15000

a) Determine marginal product function ( MP) of K and L. Determine MRTS

b) Determine Qmax

c) If he wants to produce 900 units, find out TCmin.

Answer 1

Solution:

a.). MRTS = $\frac{MP_{L} }{MP_{K} }$

Q = 2K (L – 2)

MP_L = $\frac{\partial Q} {\partial L}$ = 2K

MP_K = $\frac{\partial Q} {\partial K}$ = 2L – 4

MRTS = $\frac{2K}{2L} - 4 = \frac{K}{L} - 2$

 

b.). Qmax:

Set MRTS = $\frac{w}{r}$

w = 300

r = 600

$\frac{K}{L} - 2 =\frac{300}{600}$

K = $\frac{L}{2} - 1$

Substitute in the TC function:

TC = wL + rK

15,000 = 300L + 600K

15,000 = 300L + 600($\frac{L}{2} - 1$ )

L = 26

K = $\frac{26}{2} - 1 = 13 - 1 = 12$

Qmax (L, K) = (26, 12)

 

c.). Set MRTS = $\frac{w}{r}$

w = 300

r = 600

$\frac{K}{L} - 2 = \frac{300}{600}$

K = $\frac{L}{2} -1$

Substitute in the production function:

Q = 2K (L – 2)

900 = 2($\frac{L}{2} - 1$) (L – 2)

L = 32

K = $\frac{32}{2} - 1 = 16 - 1 = 15$

Total Cost minimum (L, K) = (32, 15)