Question

Output is produced according to Q= 4LK, where L is the quantity of labor input and K is the quantity of capital input. If the price of K is $10 and the price of Lis $5, then the cost-minimizing combination of Kand L capable of producing 32 units of output is

Answer 1

Answer :

Q = 4LK

Price of capital input = r = $10

Price of labor input = w = $5

The condition for profit optimization using optimal inputs is :

MPL / MPK = w / r

Here MPL = Marginal productivity of labor

MPK = Marginal Productivity of Capital

So : we get MPL and MPK by partially differentiating Q with respect to L and with respect to K, respectively.

MPL = 4K

Similarly MPK = 4L

So using the condition now :

4K / 4L = 5 / 10

K / L = 1 / 2

2K = L

Now output to be produced is 32 units

So Q = 4LK = 32

Also L = 2K

So, 4 * 2K * K = 32

8 * K^2 = 32

K^2 = 32 / 8 = 4

K = 2

If K = 2, then L = 2K = 2 * 2 = 4

Answer : So using 2 units of K and 4 units of L is the cost minimized combination capable of producing 32 units of output.