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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?
- jacob sLv 71 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= 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
w/r = 5/40= 1/8
Condition for maximum output:
K/L = 1/8
8K = L Equation 1
Use equation 1 in budget constraint:
L= 800/10= 80
Use L=80 in equation 1:
K= 80/8= 10
Combination of L and K that will give maximum output=(L, K)= (80, 10)
I hope this helps!