Question
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?
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?
Answers
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To determine the average product of labor (APL) function, we need to divide the total product of labor (Q) by the amount of labor input (L).
APL = Q / L = (4KL - 0.6K^2 - 0.1L^2) / L = 4K - 0.6K^2/L - 0.1L
To find the level of labor (L) at which the total output of cut-flower reaches the maximum, we need to take the derivative of the production function with respect to labor input (L) and set it equal to 0.
dQ/dL = 0 gives us 4K - 0.2L = 0
Solving for L, we get L = 20K.
Substituting L = 20K back into the production function, we can find the maximum achievable amount of cut-flower production.
Q_max = 4KL - 0.6K^2 - 0.1L^2
= 4K(20K) - 0.6K^2 - 0.1(20K)^2
= 80K^2 - 0.6K^2 - 400K^2
= 79.4K^2
So, the maximum achievable amount of cut-flower production is 79.4K^2.
APL = Q / L = (4KL - 0.6K^2 - 0.1L^2) / L = 4K - 0.6K^2/L - 0.1L
To find the level of labor (L) at which the total output of cut-flower reaches the maximum, we need to take the derivative of the production function with respect to labor input (L) and set it equal to 0.
dQ/dL = 0 gives us 4K - 0.2L = 0
Solving for L, we get L = 20K.
Substituting L = 20K back into the production function, we can find the maximum achievable amount of cut-flower production.
Q_max = 4KL - 0.6K^2 - 0.1L^2
= 4K(20K) - 0.6K^2 - 0.1(20K)^2
= 80K^2 - 0.6K^2 - 400K^2
= 79.4K^2
So, the maximum achievable amount of cut-flower production is 79.4K^2.