A businessman uses K and L to produce X. Production function is: Q = 2K ( L – 2 )
PK = 600, PL = 300, TC = 15000
a) Determine marginal product function ( MP) of K and L. Determine MRTS
b) Determine Qmax
c) If he wants to produce 900 units, find out TCmin.
Solution:
a.). MRTS = "\\frac{MP_{L} }{MP_{K} }"
Q = 2K (L – 2)
MPL = "\\frac{\\partial Q} {\\partial L}" = 2K
MPK = "\\frac{\\partial Q} {\\partial K}" = 2L – 4
MRTS = "\\frac{2K}{2L} - 4 = \\frac{K}{L} - 2"
b.). Qmax:
Set MRTS = "\\frac{w}{r}"
w = 300
r = 600
"\\frac{K}{L} - 2 =\\frac{300}{600}"
K = "\\frac{L}{2} - 1"
Substitute in the TC function:
TC = wL + rK
15,000 = 300L + 600K
15,000 = 300L + 600("\\frac{L}{2} - 1" )
L = 26
K = "\\frac{26}{2} - 1 = 13 - 1 = 12"
Qmax (L, K) = (26, 12)
c.). Set MRTS = "\\frac{w}{r}"
w = 300
r = 600
"\\frac{K}{L} - 2 = \\frac{300}{600}"
K = "\\frac{L}{2} -1"
Substitute in the production function:
Q = 2K (L – 2)
900 = 2("\\frac{L}{2} - 1") (L – 2)
L = 32
K = "\\frac{32}{2} - 1 = 16 - 1 = 15"
Total Cost minimum (L, K) = (32, 15)
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