Question

Please help me solve this problem, a detailed presentation of each step would be greatly appreciated!

1. Find the cost minimizing amount of L and K to produce 60 units of output when w=1 and r=2 2. Using the results from question 1, if the price of Q is $10/unit, calculate your profits froproducing 60 units.

Answer 1

Q = 30L - 3L² + 60K - 3K²

(1) When Q = 60, we have

30L - 3L² + 60K - 3K² = 60

10L - L² + 20K - K² = 20.........(1)

Cost is minimized when MPL/MPK = w/r = 1/2

MPL = $\dpi{80} \partial$ Q/$\dpi{80} \partial$L = 30 - 6L

MPK = $\dpi{80} \partial$ Q/$\dpi{80} \partial$K = 60 - 6K

MPL/MPK = (30 - 6L) / (60 - 6K) = (10 - 2L) / (20 - 2K) = 1/2

20 - 4L = 20 - 2K

4L = 2K

2L = K

Substituting in (1),

10L - L² + (20 x 2L) - (2L)² = 20

10L - L² + 40L - 4L² = 20

50L - 5L² = 20

5L² - 50L + 20 = 0

L² - 10L + 4 = 0

Solving this quadratic equation using online solver,

L = 9.58 or L = 0.42

K = 2 x 9.58 = 19.16, or K = 2 x 0.42 = 0.84

Therefore possible cost minimizing (L, K) bundles are: (9.58, 19.16) or (0.42, 0.84).

Since Total cost (C) = wL + rK = L + 2K,

The lower the values of L and K, the lower the cost. Therefore,

Cost is minimized when L = 0.42 and K = 0.84.

(2) Profit = Total revenue (TR) - TC

TR = Price x Q = $10 x 60 = $600

TC = 1 x 0.42 + 2 x 0.84 = 0.42 + 1.68 = $2.1

Profit ($) = 600 - 2.1 = 597.9