Question

If output is produced according to Q = 5Lk (L is the quantity of labor and k is the quantity of capital), the price of K is $12, and the price of L is $6, then the cost minimizing combination of K and L capable of producing 4,000 units of output is:__________.

A. L = 25 and k = 32.
B. L = 30 and k = 26.67.
C. L = 10 and k = 80.
D. L = 20 and k = 40.
E. L = 40 and k = 20.

Answer 1

The option E is correct

Step-by-step explanation:

Solution

Given that:

The output manufactured to Q = 5Lk

Where L= Labor quantity

k=Capital quantity

The price of K= $12

The price of L =$6

Now,

We find the combination of both K and L that will produce 4,000 units of output.

MPL/MPK is defined as the cost minimizing combination = w/r

Thus,

MPL/MPK = D(Q)/dl = 5k

same will be done for L,

MPL/MPK = D(Q)/dk = 5L

We divide 5K and 5L

So,

5k/5L =$6/$12

k/L = 1/2

Thus,

k =L/2

Now, when we substitute the value L = 2k in Q we have the following below:

Q = 5k * (2k)

Given that Q = 4000

So,

4000=10k2

4000=k2

we divide

k =20

L = 2k = 2820

= 40

Therefore, L =40, k = 20