Q = L1/2K1/2
Total cost (C) = wL + rK = L + 4K
(a) Cost is minimized when MPL / MPK = w/r = 1/4
MPL =
Q /
L = (1/2) x (K / L)1/2
MPK =
Q /
K = (1/2) x (L / K)1/2
MPL / MPK = K / L = 1/4
L = 4K
Substituting in production function when Q = 60:
L1/2K1/2 = 60
(4K)1/2K1/2 = 60
2 x K1/2 x K1/2 = 60
K = 30
L = 4 x 30 = 120
(b) When Q = 30 and L = 4K (as before),
Substituting in production function when Q = 30:
L1/2K1/2 = 30
(4K)1/2K1/2 = 30
2 x K1/2 x K1/2 = 30
K = 15
L = 4 x 15 = 60
(c) When output level falls from Q = 60 to Q = 30, quantity of Labor (L) falls by 60 (= 120 - 60) units and capital falls by 15 (= 30 - 15) units, as shown in parts (a) and (b).
(d) When K = 30,
Q = L1/2K1/2 = 30
L1/2(30)1/2 = 30
L1/2 = 30 / [(30)1/2] = (30)1/2
L = 30
NOTE: As HOMEWORKLIB Answering guideline, first 4 parts are answered.
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1. There is a furniture manufacturer using labor (L) and capital
(K) to produce tables. Its production function is given by q=
10L^.75 K^.40. It pays a
wage of $5 per hour and rents capital at a rate of $15. The firm
wants to find the cost-minimizing bundle of inputs to produce
10,000 tables. Assume K is on the y-axis in what
follows.
Write out the firm’s cost function.
Calculate the firm’s isocost equation.
What is the slope of the...