Question

III. A production process uses two inputs, labor and capital which can be written as: Q = 5LK, MPL = 5K and MPK = 5L 'w' = $150 and 'r' = $1000. Find the least cost combination of L and K when output Q = 1000 tons per day. What is the total cost of producing 1000 tons per day using this least cost combination?

Answer 1

The least cost combination of L and K will occur when the ratio of prices of labor and capital will be equal to the marginal rate of technical substitution of one input for another.

Price ratio = P_L/P_K = w/r =150/1,000 = 0.15.

Now we have to find the combination of L and K at which MRTS=0.15

MRTS= MP_L/MP_K = 5K/5L =0.15

K=0.15L

Output is Q=1000, Therefore

5LK = 1000

5L(0.15L) = 1000

0.75L²=1000

L² =1333.33

L = = 36.51 units.

K = 0.15(36.51) = 5.48 units.

The total cost of producing 1000 tons per day using the least costt combination is:

C = P_LL + P_KK

C = 150(36.51) + 1,000(5.48)

C= $5,476.50 + $5,480

C = $10,956.50

Therefore, the total cost of producing 1000 tons per day is $10,956.50