Answer to Question #240817 in Microeconomics for tutusha

Question

Suppose the production function of a firm is given as 1/ 2 1/ 2 X  0.5L K prices of labor and capital is given as $ 5 and $ 10 respectively, and the firm has a constant cost outlay of $ 600.Find the combination of labor and capital that maximizes the firm’s output

Answer 1

The production function is represented as Q = L$\times$K or

TC = L$\times$K

Where: L represent variable inputs (Labor)

K represents fixed inputs (Capital)

$\frac 12$X = 0.5L K or X = K$\times$L

L = $ 5

K = $ 10

Output maximization is obtained by minimizing cost of production

Therefore:

A combination of minimizing capital and labor is

Marginal Rate of Technical Substitution (MRTS) = $\frac {MP_L}{MP_K}$

Marginal product of labor = $\frac{dx}{dl}$ = K

'' '' '' capital =  $\frac {dx}{dk}$= L

Therefore, the marginal rate of technical substitution is = $\frac KL$

but :

X =600 (X) will be represented by Total Cost (TC)

L = $ 5

K = $ 10

$\frac KL$= $\frac{10}{5}$

k= 2L

Substituting in the equation

$\frac 12$ TC = 0.5L$\times$K

$\frac12$ TC = 0.5L$\times$ (2L)

$\frac12$ $\times$ 600 = $L^2$

300 = $L^2$

L =   $\sqrt 300$

L = 17.32

$\therefore$

300 = 0.5$\times$ (17.32)$\times$ K

K = 34.64