Answer to Question #221189 in Microeconomics for Vickie

Question

Given the following
Q=100k0.5 L0.8
C=12000 W=400=500
Determine the quantity of labour and capital
Determine the output level

Answer 1

The given production function is as follows:

$Q = 100K^{0.5}L^{0.5}$

The wage rate (W) and rent (R) is given as:

W = 400

R = 500

The cost of production (C) is given as:

C = 12000

The firm produces at the level at which marginal rate of technical substitution (MRTS) is equal to the price ratio.

MRTS is the ratio of marginal products of labor and capital:

$MRTS=\frac{MP_L}{MP_K}$

$=\frac{\frac{\delta Q}{\delta L}}{\frac{\delta Q}{\delta K}}$

$=\frac{\frac{0.8(100)K^{0.5}}{L^{0.2}}}{\frac{0.5(100)L^{0.8}}{K^{0.5}}}$

$=\frac{8K}{5L}$

Equate MRTS to price ratio :

$MRTS=\frac{W}{R}$

$\frac{8K}{5L}=\frac{400}{500}$

$4000K=2000L$

$K=\frac{1}{2}L(1)$

The cost constraint is as follows:

$C=WL+RK\\12000=400L+500(\frac{1}{2}L)\\12000=650L\\L=18(approx)$

Substitute value of L in equation 1 to determine K:

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

$=\frac{1}{2}(18)$

$=9$

The quantity (Q) is determined as follows:

$Q=100K^{0.5}L^{0.8}\\=100(9)^{0.5}(18)^{0.8}\\=10(3)(10.09)\\=3027$

Thus, the value of Q is 3027.