Suppose the short-run production function of good A is given by Q = 8+4LK-5L^2+0.8K^2, where Q is quantity of good A produced, L is labour input and K is fixed capital input (k = 10).
A)Determine the product of labour (APL) function?
B)At what level of labour does the total output of good A reach maximum?
C)What will be the maximum production of good A?
To determine the product of labour (APL) function, divide the total output (Q) by the labor input (L):
APL = Q / L = (8 + 4LK - 5L^2 + 0.8K^2) / L
To find the level of labor where the total output reaches its maximum, we need to take the derivative of the production function with respect to labor (L) and set it equal to zero:
∂Q/∂L = 4K - 10L + 0.8K^2 = 0
Simplifying the equation:
10L = 4K + 0.8K^2
Substituting the value of K=10:
10L = 4(10) + 0.8(10)^2
10L = 40 + 80
10L = 120
L = 12
Therefore, the level of labor where the total output reaches its maximum is 12.
To find the maximum production of good A, substitute the value of L=12 into the production function:
Q = 8 + 4(12)(10) - 5(12)^2 + 0.8(10)^2
Q = 8 + 480 - 5(144) + 80
Q = 8 + 480 - 720 + 80
Q = -152
Therefore, the maximum production of good A is -152 units.
A) The average product of labor (APL) is calculated by dividing the total product of labor (TPL) by the quantity of labor input (L). To find the APL, we can differentiate the production function with respect to labor input and then substitute the given values of K and k into the resulting expression.
Q = 8 + 4LK - 5L^2 + 0.8K^2
Differentiating with respect to L:
dQ/dL = 4K - 10L
Substituting K = k = 10:
dQ/dL = 4(10) - 10L
= 40 - 10L
The APL function is derived by dividing the total product function by the labor input function:
APL = TPL / L
Substituting Q = TPL and differentiating with respect to L:
APL = (dQ/dL) / 1
APL = (40 - 10L) / 1
APL = 40 - 10L
Therefore, the product of labor (APL) function is APL = 40 - 10L.
B) To find the level of labor at which the total output (Q) reaches its maximum, we need to find the point where the marginal product of labor (MPL) equals zero. The MPL is calculated by differentiating the production function with respect to labor input.
Q = 8 + 4LK - 5L^2 + 0.8K^2
Differentiating with respect to L:
dQ/dL = 4K - 10L
Setting MPL equal to zero:
4K - 10L = 0
10L = 4K
L = (4K) / 10
L = (4(10)) / 10
L = 4
At a labor input level of 4, the total output of good A reaches its maximum.
C) To find the maximum production of good A, we substitute the value of L into the production function.
Q = 8 + 4LK - 5L^2 + 0.8K^2
Substituting L = 4 and K = 10:
Q = 8 + 4(10)(4) - 5(4)^2 + 0.8(10)^2
Q = 8 + 160 - 80 + 80
Q = 168
Therefore, the maximum production of good A is 168.
To determine the average product of labor (APL) function, we need to calculate the output per unit of labor input.
The formula for average product of labor (APL) is:
APL = Q / L
where Q is the quantity of good A produced and L is the labor input.
Using the given production function Q = 8 + 4LK - 5L^2 + 0.8K^2 and assuming K = 10, we can calculate APL.
A) Determining the average product of labor (APL) function:
APL = Q / L
Substituting Q = 8 + 4LK - 5L^2 + 0.8K^2 and K = 10:
APL = (8 + 4L(10) - 5L^2 + 0.8(10)^2) / L
Simplifying the equation:
APL = (8 + 40L - 5L^2 + 80) / L
APL = (88 + 40L - 5L^2) / L
Thus, the average product of labor (APL) function is given by APL = (88 + 40L - 5L^2) / L.
B) To find the level of labor at which the total output of good A reaches maximum, we need to find the maximum point on the production function.
Differentiate the production function with respect to labor (L) and set it equal to zero to find the critical point:
dQ/dL = 0
Using the production function Q = 8 + 4LK - 5L^2 + 0.8K^2 and K = 10:
dQ/dL = 4K - 10L = 0
4(10) - 10L = 0
40 - 10L = 0
10L = 40
L = 4
So, at L = 4 (units of labor), the total output of good A reaches its maximum.
C) To find the maximum production of good A, substitute the value of L in the production function:
Q = 8 + 4LK - 5L^2 + 0.8K^2
Q = 8 + 4(10)(4) - 5(4)^2 + 0.8(10)^2
Q = 8 + 160 - 80 + 80
Q = 168
Therefore, the maximum production of good A is 168 units.