Question

The production function for global electronics is Q=2k^0.5 L^0.5

Assume that the capital stock is fixed at nine units (i.e., K = 9). If the price of output (P) is Rs.6 per unit and the wage rate (w) is Rs.2 per unit, determine the optimal or profit maximizing rate of labor to be hired. What labor rate is optimal if the wage rate increased to Rs.3 per unit?

Answer 1

Profit=Total Revenue(TR)- Total cost (TC)

Total Revenue=Price×Quantity

Total Cost=wL

$K=9, Q=2(9)^{0.5}L^{0.5}=2(3)L^{0.5}=6L^{0.5}$

$Profit: π=6×(6L^{0.5})−2L\\\pi=36L^{0.5}−2L\\\frac{d\pi}{dL}=0.5(36)L^{−0.5}−2\\=18L^{−0.5}−2\\$

equiting $\frac{d\pi}{dL}=0$ we get

$18L^{-0.5}-2=0\\L^{-0.5}=\frac{2}{18}\\L^{0.5}=9$

squaring both sides, we get

L=81

Therefore, Profit maximizing rate of Labor to be hired is 81.

When w=3, we get

$Profit: π=6×(6L^{0.5})−3L\\\pi=36L^{0.5}−3L\\\frac{d\pi}{dL}=0.5(36)L^{−0.5}−3\\=18L^{−0.5}−3\\$

equiting $\frac{d\pi}{dL}=0$ we get

$18L^{-0.5}-3=0\\L^{-0.5}=\frac{3}{18}\\L^{0.5}=6$

squaring both sides, we get

L=36

The rate of labor that is optimal if the wage rate is Rs. 2 per unit is 36.