Question

A firm production is represented by the following Cobb-Douglas function: Q = K^1/5 L^4/5. The rental rate, r, of capital is given by $240 and the wage rate is $30.
a. For a given level of output, what should be the ratio of capital to labor in order to minimize
costs?
b. How much capital and labor should be used to produce 400 units? How much is the total cost?
c. What is the short run total cost if output is decreased to 300 units?
d. How would the capital labor choice and total cost would change in the long run?
e. Does this production function exhibit increasing, decreasing, or constant returns to scale?
Please answer based on the cost calculations in parts b and d.

Answer 1

a) Cost is minimized when MRTS = w/r
MRTS = MPL/MPK = (∂Q/∂L)/(∂Q/∂K) = (4/5)K^1/5L^(4/5)-1/(1/5)K^(1/5)-1L^4/5 = 4K^1/5L^-1/5/K^-4/5L^4/5
= 4K^(1/5)+(4/5)/L^(4/5)+(1/5) = 4K/L
So, 4K/L = 30/240 = 1/8
So, K/L = 1/8*4
So, K/L = 1/32

b) So, L = 32K
Q = K^1/5 L^4/5 = 400
So, K^0.2(32K)^0.8 = (32)^0.8(K)^0.2+0.8 = 400
So, K = 400/32^0.8 = 400/16
So, K = 25
L = 32*25
So, L = 800
TC = wL + rK = 30*800 + 240*25 = 24,000 + 6,000 = 30,000
So, TC = 30,000

c) In short run, K = 25
So, K^0.2L^0.8 = 300
So, L^0.8 = 300/(25)^0.2 = 300/1.91 = 157.07
So, L = (157.07)^1/0.8
So, L = 556.05

SRTC = wL + rK = 30*556.05 + 240*25 = 16,681.5 + 6,000 = 22,681.5
So, TC = 22,681.5

d. Q = K^1/5 L^4/5 = 400
So, K^0.2(32K)^0.8 = (32)^0.8(K)^0.2+0.8 = 300
So, K = 300/32^0.8 = 300/16
So, K = 18.75
L = 32*18.75
So, L = 600
TC = wL + rK = 30*600 + 240*18.75 = 18,000 + 4500 = 22,500
So, TC = 22,500

So, total cost would decrease, L increases while L decrease.

e. Let K = tK and L = tL where t > 1.
So, Q' = (tK)^1/5 (tL)^4/5 = (t)^(1/5)+(4/5)K^1/5 L^4/5 = tQ
So, there is constant returns to scale as power of t is 1.