Question

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

Input limit: K + L = 100
Budget limit: 8 x K + 10 x L = 840

b. What are the quantities of two inputs the firm must buy in order to produce a maximum output, given input and budget constraints?

L=
K=

Answer 1

Answer:

L= 80

K= 20

Explanation:

The guy who solved this question he perfectly explained but he did mistake on putting a correct answer at L=20 and K=80. Just reverse the L and K mistake.

Answer 2

Answer:

a.

Input limit: $K+L=100$

Budget limit: $8K+10L=840$

b.

L=20

K=80

Explanation:

Ok, you save me a little bit of work and resolved the point a.

Indeed, the equations for Input limit and Budget limit are the ones you put in the question.

Now I will show you how to find the quantities of the two inputs in order to have a maximum output taking into account the input limit and the budget limit:

First, we need to put our equations in function of one of the variables, lets do it with L (Labor):

Input limit:

$K+L=100$

$L=100-K$

Budget Limit:

$8K+10L=840$

$L=\frac{840-8K}{10}$

$L=84-0.8K$

Now we match the 2 equations and find the value of K

L=L

$100-K=84-0.8K$

$100-84=-0.8K + K$

$0.2K=16$

$K=\frac{16}{0.2}$

$K=80$

Now that we have the optimum K we replace in any of the two equations to find the optimum L

$L=100-80=20$

And then we replace in the Input limit and Budget limit equations to verify:

Input limit:

$80 + 20 = 100$

Budget Limit:

$8(80)+10(20)=840$