Question

1.    Given the following production function Q = KL² and the unit price of labor and capital to be Birr 15 and 10 respectively, then

a.   What combination of labor and capital minimizes the cost of producing 384 units?

b.   What is the minimum cost?

c .Show the cost minimizing condition graphically?

Answer 1

$10K+15L-\lambda(KL^2-384)\\10-\lambda(L^2)=0\\15-\lambda(2KL)=0\\\frac{10}{2KL}=\frac{15}{L^2}\\\frac{15}{10}=\frac{L^2}{2KL}\\3=\frac{L}{K}\\L=3K\\384=K(3K)^2\\384=3K^3\\128=K^3\\K=5.03968$

$384=KL^2\\384=\frac{L}{3}×L^ 2\\384×3=L^3\\1152=L^3\\L=10.4829$

1: The combination of labor and capital minimizes the cost of producing 384 units

$K=5.03968\\L=10.4829$

2:The minimum cost

$5.03968×10+10.4829×15\\207.6403$

3: