Question

A firm buys two inputs, labor L and capital K, the total amount of which cannot exceed 100 units. The wage is $8, and the rental rate of capital is $10. The firm can, at most, spend $840 on the two inputs. This data can be summarized in the following equations:

Input limit. K + L = 100
Budget limit: 8L + 10K = 840
What are the quantities of two inputs the firm must buy in order to produce a maximum output, given input and budget constraints?
LE=
K=

Answer 1

a.

Input limit: K + L = 100

Budget limit: 8L + 10K = 840

b.

L = 80

K = 20

See below

Step-by-step explanation:

With regards to the above, we need to develop equations and put in function of one of the variable , say with L(labor)

Input limit

K + L = 100. Equation 1

Budget limit

8L + 10K = 840 Equation 2

From equation 1, make K subject of the formula

K + L = 100

K = 100 - L

Substitute for K in equation 2

8L + 10K = 840

8L + 10(100 - L) = 840

8L + 1,000 - 10L = 840

8L - 10L = 840 - 1,000

-2L = -160

L = 80

We can also get the value of K by substituting for L in equation 1.

K + L = 100

K + 80 = 100

K = 100 - 80

K = 20

Since we have the optimum K, we can replace in any of two equations to find the optimum L,

Also, we will replace the input limit and budget limit equation to verify

Input limit

K + L = 100

20 + 80 = 100

Budget limit

8L + 10K = 840

8(80) + 10(20) = 840

640 + 200 = 840

840 = 840