Q = K1/2L1/2 w = $2, r = $2 The firm would like to know the maximum ou

Q = K1/2L1/2 w = $2, r = $2 The firm would like to know the maximum output that can be produced for $8,000.

Find the combination of inputs that maximizes output for a cost of $8,000, the amount of output that can be produced, and identify the expansion path.

ANSWER

MPL = 1/2 K1/2L-1/2
MPK = 1/2 K-1/2L1/2
Optimization requires:
MPL/w = MPK/r
This results in K=L, which is the equation of the expansion path.
TC = 8,000 = 2L + 2K
Substitute the expansion path relationship to yield:
K* = L* = 2000
Then Q = K1/2L1/2 = 2000.