Question

. A firm faces following production function: ?? = ?? 1 2?? 1 2. Suppose the rental rate of capital r=40 and wage rate for labor w=10. (25) a) For a given level of output, what should be the optimal ratio of capital to labor in order to minimize cost? b) What is the minimum cost of producing 200 units? At minimum cost for producing 200 units, how much capital and labor are needed? c) What is the minimum cost of producing 300 units? At minimum cost for producing 300 units, how much capital and labor are needed? d) Graphically show the long run-expansion path. e) Does this production function exhibit increasing, decreasing, or constant returns to scale?

Answer 1

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

Total cost (TC) = wL + rK = 10L + 40K

(a) Cost is minimized when MPL/MPK = w/r = 10/40 = 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

(b) K/L = 1/4, therefore L = 4K

When Q = 200,

K^1/2L^1/2 = 200

Squaring both sides,

KL = 40,000

K x 4K = 40,000

4K² = 40,000

K² = 10,000

K = 100

L = 4 x 100 = 400

TC = 10 x 400 + 40 x 100 = 4,000 + 4,000 = 8,000

(c) K/L = 1/4, therefore L = 4K

When Q = 300,

K^1/2L^1/2 = 300

Squaring both sides,

KL = 90,000

K x 4K = 90,000

4K² = 90,000

K² = 22,500

K = 150

L = 4 x 150 = 600

TC = 10 x 600 + 40 x 150 = 6,000 + 6,000 = 12,000

(d)

When Q = 200, K = 100 and L = 400 while when Q = 300, K = 150 and L = 600. The expansion path is the line connecting points A and B in following graph.

(e) Let us double both inputs so that new production function becomes

Q1 = (2K)^1/2(2L)^1/2 = 2^1/22^1/2K^1/2L^1/2 = 2 x K^1/2L^1/2 = 2Q

Q1/Q = 2

Since doubling both inputs exactly doubles output, there is constant returns to scale.