Example: Suppose that the short-run production function of certain cut-flower firm is given by:
𝑸=𝟒𝑲𝑳−𝟎.𝟔𝑲^𝟐-0.1𝑳^𝟐 :where Q is quantity of cut-flower produced, L is labour input and K is fixed capital input (K=5).
Determine the average product of labour (APL) function.
At what level of labour does the total output of cut-flower reach the maximum?
What will be the maximum achievable amount of cut-flower production?
To determine the average product of labour (APL) function, we need to divide the total product (Q) by the amount of labor input (L).
APL = Q / L
Given the production function Q = 4KL - 0.6K^2 - 0.1L^2, we substitute K=5 for the fixed capital input:
Q = 4(5)L - 0.6(5)^2 - 0.1L^2
Q = 20L - 15 - 0.1L^2
Now we can calculate APL:
APL = (20L - 15 - 0.1L^2) / L
APL = 20 - (15/L) - 0.1L
To find the level of labor at which the total output of cut-flower reaches the maximum, we need to find the maximum point of the production function. We can do this by taking the derivative of Q with respect to L and setting it equal to zero.
dQ/dL = 20 - (15/L) - 0.1L
0 = -15/L - 0.1
0.1 = -15/L
1/L = 0.1/15
L = 15/0.1
L = 150
When L = 150, the total output of cut-flower reaches its maximum.
To find the maximum achievable amount of cut-flower production, we substitute L = 150 into the production function:
Q = 4(5)(150) - 0.6(5)^2 - 0.1(150)^2
Q = 3000 - 15 - 2250
Q = 735
The maximum achievable amount of cut-flower production is 735.
To find the average product of labor (APL) function, we need to divide the total output (Q) by the labor input (L).
1. Average Product of Labor (APL) Function:
APL = Q / L
Given the short-run production function: Q = 4KL - 0.6K^2 - 0.1L^2
To find the APL function, we substitute this expression for Q:
APL = (4KL - 0.6K^2 - 0.1L^2) / L
Simplifying, we get:
APL = 4K - 0.6K^2/L - 0.1L
2. Level of Labor for Maximum Total Output:
To find the level of labor at which the total output of cut-flowers reaches its maximum, we need to find the maximum point on the production function.
Given the production function: Q = 4KL - 0.6K^2 - 0.1L^2
To find the maximum, we take the derivative of the production function with respect to labor (L) and set it equal to zero:
dQ/dL = 4K - 0.2L = 0
Solving for L, we get:
4K = 0.2L
L = 20K
Therefore, the level of labor at which the total output of cut-flowers reaches its maximum is L = 20K.
3. Maximum Achievable Cut-Flower Production:
To find the maximum achievable amount of cut-flower production, we substitute the value of L obtained above into the production function:
Q = 4KL - 0.6K^2 - 0.1L^2
Q = 4K(20K) - 0.6K^2 - 0.1(20K)^2
Q = 80K^2 - 0.6K^2 - 0.1(400K^2)
Q = 80K^2 - 0.6K^2 - 40K^2
Q = 39.4K^2
Therefore, the maximum achievable amount of cut-flower production is given by the function Q = 39.4K^2.