Assume that the annual production function for good A is given by Q= 4k^1/2Ld^1/2L^1/2 , where k= capital , Ld = land, L= labour . When the assumption that K = 16 and Ld = 64, determine:

Question

A. Average product of 4 units of labour
B. Marginal product of labour when labour equals 4 units

Answer 1

To find the average product of 4 units of labor (L), we need to divide the total product (Q) at that level of labor by the number of units of labor:

A. Average product of 4 units of labor:
Average product = Q / L

Given: Q = 4k^(1/2)Ld^(1/2)L^(1/2), k = 16, Ld = 64, and L = 4

Average product = 4(16)^(1/2)(64)^(1/2)(4)^(1/2) / 4
= 4(4)(8)(2) / 4
= 256

Therefore, the average product of 4 units of labor is 256.

To find the marginal product of labor (L) when labor equals 4 units, we need to find the change in total product (Q) when labor increases by 1 unit:

B. Marginal product of labor when labor equals 4 units (MP):
Marginal product = ΔQ / ΔL

Given: Q = 4k^(1/2)Ld^(1/2)L^(1/2), k = 16, Ld = 64, and L = 4

To find the change in total product, we need to evaluate Q when L = 4 and L = 5:
ΔQ = Q(L = 5) - Q(L = 4)
= 4(16)^(1/2)(64)^(1/2)(5)^(1/2) - 4(16)^(1/2)(64)^(1/2)(4)^(1/2)
= 4(4)(8)(√5) - 4(4)(8)(√4)
= 16(8)(√5) - 16(8)(2)
= 128(√5 - 2)

To find the change in labor, we subtract the initial level of labor from the final level of labor:
ΔL = L - (L = 4)
= 5 - 4
= 1

Marginal product = (128(√5 - 2)) / 1
= 128(√5 - 2)

Therefore, the marginal product of labor when labor equals 4 units is 128(√5 - 2).