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# please any body who can help me ?

A firm faces the production function Q = 12K0.4L0.4 and can buy the inputs K and L at prices per unit of ETB40 and ETB5 respectively. If it has a budget of ETB800. What combination of K and L should it use in order to produce the maximum possible output? What do the first order conditions inform about the inputs?

Verify that this is a maximum?

### 1 Answer

Relevance
• Q= 12K0.4 L0.4

Cost of labor=w= \$5

Cost capital=r = \$40

Total budget= \$800

Budget constraint: 5L+40K= 800

For Maximizing output condition:

MRTS= w/r

MRTS= Marginal product of L(MPL)/Marginal product of capital(MPK)

MPL= Differentiation of Q with respect to L= 12(0.4)K0.4 L-0.6

MPK= Differentiation of Q with respect to K= 12(0.4)K-0.6 L0.4

MRTS= K/L

w/r = 5/40= 1/8

Condition for maximum output:

K/L = 1/8

8K = L Equation 1

Use equation 1 in budget constraint:

5L+40K= 800

5L+5L= 800

10L= 800

L= 800/10= 80

Use L=80 in equation 1:

8K= L

K= 80/8= 10

Combination of L and K that will give maximum output=(L, K)= (80, 10)

I hope this helps!

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