Question

1. Using the Cobb-Douglas production function:

Y_{t} =
A_{t}K_{t}^{1/3}L_{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} = L^{2} x 900

L^{2} = (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.

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