Question

1. Using the Cobb-Douglas production function:

Y_t = A_tK_t^1/3L_t^2/3

If K = 27, L = 8 A = 2, and α = 1/3, what is the value of Y? (For K and L, round to the nearest whole number) ______

2. If Y = 300, L = 10, and α = 1/3, what is the marginal product of labor? ______

3. Using the values for Y and α above, if K = 900, what is the marginal product of capital? Express your answer as a percentage. _______

4. For the Cobb-Douglas production function, if α = 1/5, what would labor’s share of output be?________

5. For the Cobb-Douglas production function, if A = 2 and Y =100, what would Y be if A increases to a value of 3?

______

Answer 1

(1) Plugging in given values,

Y = 2 x (27)^1/3 x (8)^2/3 = 2 x 3 x 4 = 24

(2) Plugging in given values,

300 = 2 x (K)^1/3 x (10)^2/3

150 = (K)^1/3 x (10)^2/3

Taking (3/2)-th root on both sides,

(150)^3/2 = (K)^1/2 x 10

Squaring,

(150)³ = K x 100

K = (150 x 150 x 150) / 100 = 33,750

MPL = Y/L = 2 x (2/3) x (K/L)^1/3 = (4/3) x (33,750 / 10)^1/3 = (4/3) x (3,375)^1/3 = (4/3) x 15 = 20

(3) Plugging in given values,

300 = 2 x (900)^1/3 x (L)^2/3

150 = (900)^1/3 x (L)^2/3

Taking (3/2)-th root on both sides,

(150)^3/2 = (900)^1/2 x L

Squaring,

(150)³ = L² x 900

L² = (150 x 150 x 150) / 900 = 3,750

L = 61.24

MPK = Y/K = 2 x (1/3) x (L/K)^2/3 = (2/3) x (61.24 / 900)^2/3 = (2/3) x (0.068)^2/3 = (2/3) x 0.1667 = 0.1111 = 11.11%

NOTE: As per Answering Policy, first 3 questions are answered.