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:
A. Average product of 4 units of labour
B. Marginal product of labour when labour equals 4 units
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).
To determine the average product of 4 units of labor (A), and the marginal product of labor when labor equals 4 units (B), we need to substitute the given values into the production function.
A. Average product of 4 units of labor (A):
The average product of labor is calculated by dividing the total product by the amount of labor. In this case, we need to calculate the total product (Q) using the given values of capital (k = 16), land (Ld = 64), and labor (L = 4).
Substituting the values into the production function:
Q = 4k^(1/2) * Ld^(1/2) * L^(1/2)
= 4 * 16^(1/2) * 64^(1/2) * 4^(1/2)
To simplify the calculation, we can first compute the square root of each value separately:
16^(1/2) = √16 = 4
64^(1/2) = √64 = 8
4^(1/2) = √4 = 2
Now, substitute the simplified values back into the equation:
Q = 4 * 4 * 8 * 2 = 256
The total product is equal to 256.
Next, divide the total product by the amount of labor to determine the average product of labor:
Average Product of Labor = Q / L
= 256 / 4
= 64
Therefore, the average product of 4 units of labor is 64.
B. Marginal product of labor when labor equals 4 units (B):
The marginal product of labor measures the change in total product resulting from a change in labor input. To calculate it, we need to determine the total product (Q) when the labor input is 4 units, and then subtract the total product when the labor input is 3 units.
Substituting the values into the production function:
Q = 4 * 16^(1/2) * 64^(1/2) * 4^(1/2)
= 4 * 4 * 8 * 2
= 256
Now, let's calculate the total product when the labor input is 3 units:
Q' = 4 * 4 * 8 * 3^(1/2)
= 256
The marginal product of labor is given by the difference between the total product for 4 units of labor (Q) and the total product for 3 units of labor (Q'):
Marginal Product of Labor = Q - Q'
= 256 - 256
= 0
Therefore, the marginal product of labor when labor equals 4 units is 0.
To determine the average product of 4 units of labor, we need to find the total product of labor when labor equals 4 units and divide it by 4.
Step 1: Substitute the given values into the production function:
Q = 4k^(1/2)Ld^(1/2)L^(1/2)
Q = 4(16)^(1/2)(64)^(1/2)(4)^(1/2)
Step 2: Simplify the expression:
Q = 4(4)(8)(2)
Q = 256
Step 3: Calculate the average product of labor:
Average product = Total product / Units of labor
Average product = 256 / 4
Average product = 64 units of output
Therefore, the average product of 4 units of labor is 64 units of output.
To determine the marginal product of labor when labor equals 4 units, we need to find the change in total product when labor is increased by 1 unit.
Step 1: Substitute the given values into the production function:
Q = 4k^(1/2)Ld^(1/2)L^(1/2)
Q = 4(16)^(1/2)(64)^(1/2)(4)^(1/2)
Step 2: Simplify the expression:
Q = 4(4)(8)(2)
Q = 256
Step 3: Find the total product when labor equals 4 units:
Total product = Q = 256
Step 4: Increase the units of labor by 1 and calculate the new total product:
Q' = 4(4)(8)(3)^(1/2)
Q' = 4(4)(8)(√3)
Step 5: Calculate the marginal product of labor:
Marginal product = Change in total product / Change in labor
Marginal product = Q' - Q / 1
Marginal product = (4(4)(8)(√3)) - 256 / 1
Marginal product = (128√3) - 256
Therefore, the marginal product of labor when labor equals 4 units is (128√3) - 256.