Answer to Question #267650 in Microeconomics for Katie

Question #267650

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.


1
Expert's answer
2021-11-19T11:11:18-0500

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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