Question

Producer

Suppose

Firm Production Function Q=50 à ƒ ƒ ¢ ˆ šLK

MPLE 25 VE MPK = 25à ƒ ƒ ¢ ˆ š2/

suppose

Price of Labor =W= P5.

Price of Capital = t = 120

Fand the cost minimizing input Combination if the firm wants to produce

1000 units.

Q=1600

Answer 1

Cost minimizing combination of capital and labour is one where the marginal rate of substitution(MRTS) is equal to:

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

Marginal product of labour is $\frac{\delta Q}{\delta L}$ =50 Ã ¢ K

Marginal product of capital= $\frac{\delta Q}{\delta K}$ =50 Ã ¢ L

Therefore MRTS=$\frac{50 Ã ¢ K}{50 Ã ¢ L}=\frac{K}{L}$

Set the marginal rate of technical substitution equal to the input price ratio to

determine the optimal capital-labor ratio.

$\frac{K}{L}=\frac{P5}{120} \therefore K=\frac{5PL}{120}$ and $L=\frac{120K}{5P}$

Substitute for L in the production function and solve where K yields an output of 1000 units.

$1000=50\times\frac{120K}{5P}\times K$

$200=\frac{120K^{2}}{5P}$

$1000P=120K^{2}$

$K^{2}=8.33P \therefore K=(8.33P)^{0.5}$

To get L,we substitute K above in $L=\frac{120K}{5P}$

$L=\frac{120K}{5P}\times (8.33P)^{0.5}$

$L=\frac {24(8.33P)^{0.5}}{P}$

$\therefore K=(8.33P)^{0.5}$ and $L=\frac{24(8.33P)^{0.5}}{P}$ are the cost-minimizing levels of K and L.