Question

instructions- You have to use Python in order to do this Problem Set.

4 Cobb-Douglas Production Function Suppose a firm uses the following production function Y = z × K0.3 × N 0.7 d , where Y is output, z is TFP, K is capital, and Nd is labor. 1. Set z = 10, K = 5 and plot the production function over the range Nd ∈ [0, 5] 2. Derive this function w.r.t. Nd.Set z = 10, K = 5, and plot the marginal product of labor (MPN) over the range Nd ∈ [0, 5] 3. What does the MPN measure. Explain in words.

Answer 1

1)

Y=zK^0.3N_d^0.7

Set z=10 and K=5

Y=10*5^0.3N_d^0.7

Y=16.20657*N_d^0.7

Following output table can be made for Nd ∈ [0, 5]

Nd	Y=16.20657*Nd0.7
0	0.00000
1	16.20657
2	22.91955
3	28.07060
4	32.41313
5	36.23898

We can draw output function in the given range of N_d as under

2)

Y=zK^0.3N_d^0.7

Marginal product of N_d can be found by differentiate Y with respect to N_d, we get

Set z=10, K=5, we get

Following output table can be made for Nd ∈ [0, 5]

Nd	Y=11.34460/Nd0.3
0.1	22.63545
1	11.34460
2	9.21468
3	8.15930
4	7.48464
5	7.00000

MPN represents the marginal product of labor. It means it is the change in output resulting from employing one more labor.

We find that MPN is decreasing for increasing value of Nd. It means that adding more units of labor is resulting in the smaller change in output.