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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

  • 1 month ago

    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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