Question

2. A firm produces a product with labor and capital. Its production function is described by Q = L1/2K1/2 The price of labor is w = 1, while that of capital is r = 4. (a) What is the cost-minimizing input bundle when Q = 60? (b) What is the cost-minimizing input bundle when = 30? (c) The desired output level falls from -60 to Q = 30, what is the new long-run cost-minimizing input bundle? (d) In the short-run, K = 30 and cannot be changed. In this case, what is the optimal short-run labor input? (e) Find the lonug-run total cost function. (f) Sketch the long-run total cost function and interpret its slope.

Answer 1

Q = L^1/2K^1/2

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:

L^1/2K^1/2 = 60

(4K)^1/2K^1/2 = 60

2 x K^1/2 x K^1/2 = 60

K = 30

L = 4 x 30 = 120

(b) When Q = 30 and L = 4K (as before),

Substituting in production function when Q = 30:

L^1/2K^1/2 = 30

(4K)^1/2K^1/2 = 30

2 x K^1/2 x K^1/2 = 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 = L^1/2K^1/2 = 30

L^1/2(30)^1/2 = 30

L^1/2 = 30 / [(30)^1/2] = (30)^1/2

L = 30

NOTE: As HOMEWORKLIB Answering guideline, first 4 parts are answered.