Question

The production function is q=L^0.6K ^0.4 .

1. The company must produce 15 units. The cost of capital is $10, and the cost of labor is $5. Using the Lagrangian multiplier, calculate the combination of labor and capital, K* and L*, where you can minimize the cost of producing 15 units.

Answer 1

First, let us write the firm's problem. min Te - 5L + 10k (LR) (As, the cost of labour -25) (d, the cost of capital - $10 which is subject to a a = (L) 0.6 (K) 0.4 15 (4) 06 (6) 04 – (As a = 15 units) Therefore, selling up the lagrangian Sunction we get 0 = (5L +20k) + (15 – 106 40.4) where, a= lagranglan гоманоlun 1

Noes, partially differentiating the lagrangian with respect to , and @ & also equating the resulting derivatives to zero we get the necessary conditions for a minimum. Here of 20 - 2 (0.4) (4) 0 - 4 22 06_07 20 = 5.-- (06)(30.4 (1)*0.450 1 152 (4) 06 (6) 04

from the top (2) canation we get - a co-c) (130:403-04 2 (0:4) (K)-0.6 (4) 0.6 ou, no ne (1) on, L = () 04 ķ = 6) = 0.33 rka 0.33L

Cutting sko.B3L in canation (4) we get 15- 2016/04 0% (L.)046 (1-3004 = 15 oy (L.) 0:(0.231) 04 = 15 O = 15 (0.33 0.9 23.43 norit we know, K = 0.33L or ke 0.33 X 23-43 - t. 73 units ? Therefore, the combination of labor and capital kt and it where we can minimize the cost of producing 15 units will be - * 20.40 wouds. 4 K - fit3 units