5.Suppose that the short-run production function for cut-flower by a certain Ethiopian firm is given by: Q= -0.2L2 -1.2K2 +8LK
Where Q - represents the annual quantity of cut-flower produced.
K - annual capital input; suppose K=10.
L - annual labor input.
a.Determine the average product of labor (APL) function.
b.At what level of labor does the total output of cut-flower reach the maximum?
c.What will be the maximum achievable amount of cut-flower production?
a. To find the average product of labor (APL), we need to divide the total output (Q) by the labor input (L).
APL = Q/L
Substituting the given production function, we have:
APL = (-0.2L^2 - 1.2K^2 + 8LK) / L
Since K is given as 10, we can simplify further:
APL = (-0.2L^2 - 1.2(10)^2 + 8L(10)) / L
= -0.2L^2 - 120 + 80L / L
= -0.2L - 120 + 80
= -0.2L - 40
Therefore, the average product of labor (APL) function is APL = -0.2L - 40.
b. To find the level of labor at which total output of cut-flower reaches the maximum, we need to find the point where the marginal product of labor (MPL) equals zero.
The marginal product of labor (MPL) is the derivative of the production function with respect to labor (L):
MPL = dQ/dL = -0.4L + 8K
Substituting K = 10:
MPL = -0.4L + 8(10)
= -0.4L + 80
Setting MPL equal to zero:
-0.4L + 80 = 0
-0.4L = -80
L = (-80) / (-0.4)
L = 200
Therefore, at a labor input of L = 200, the total output of cut-flower reaches its maximum.
c. To find the maximum achievable amount of cut-flower production, we substitute the value of L = 200 into the production function:
Q = -0.2L^2 - 1.2K^2 + 8LK
= -0.2(200)^2 - 1.2(10)^2 + 8(200)(10)
= -0.2(40000) - 1.2(100) + 8(2000)
= -8000 - 120 + 16000
= 8000 - 120
= 7880
Therefore, the maximum achievable amount of cut-flower production is 7880 units.
a. To determine the average product of labor (APL), we need to divide the total product (Q) by the amount of labor (L).
The total product (Q) is given by the production function Q = -0.2L^2 - 1.2K^2 + 8LK.
Therefore, the average product of labor (APL) can be calculated as APL = (Q / L) = (-0.2L^2 - 1.2K^2 + 8LK) / L.
Simplifying this equation, we have APL = -0.2L - 1.2K + 8K.
Since the value of K is given as K = 10, we substitute this value into the equation to get the final expression for APL: APL = -0.2L - 1.2(10) + 8(10).
b. To find the level of labor at which the total output of cut-flower reaches the maximum, we need to find the point where the marginal product of labor (MPL) equals zero.
The marginal product of labor (MPL) is the derivative of the total product function with respect to labor (L). Let's differentiate the production function Q = -0.2L^2 - 1.2K^2 + 8LK with respect to L:
∂Q/∂L = -0.4L + 8K.
Setting this equation equal to zero, we have: -0.4L + 8K = 0.
Substituting the given value K = 10, we get: -0.4L + 8(10) = 0.
Simplifying this equation: -0.4L + 80 = 0, we can solve for L:
-0.4L = -80,
L = -80 / -0.4,
L = 200.
Therefore, the level of labor at which the total output reaches the maximum is L = 200.
c. To find the maximum achievable amount of cut-flower production, we substitute this value of L=200 into the production function:
Q = -0.2(200)^2 - 1.2(10)^2 + 8(200)(10).
Calculating this equation will give us the maximum achievable amount of cut-flower production.
a. To determine the average product of labor (APL) function, we need to calculate the total output of cut-flower for each level of labor input and divide it by the corresponding level of labor input.
Given the production function: Q = -0.2L^2 - 1.2K^2 + 8LK, and K = 10 (constant), we can now calculate APL.
APL = Q / L
Substituting the given values, we have:
APL = (-0.2L^2 - 1.2K^2 + 8LK) / L
Since K is constant (K = 10), we can simplify the equation further:
APL = (-0.2L^2 + 80L) / L
b. To find the level of labor at which the total output of cut-flower reaches its maximum, we need to differentiate the production function with respect to labor (L) and set it equal to zero.
So, first, let's differentiate the production function:
dQ/dL = -0.4L + 8K
Substituting the given constant value of K (K = 10) into the equation:
dQ/dL = -0.4L + 80
To maximize Q, set dQ/dL equal to zero and solve for L:
-0.4L + 80 = 0
-0.4L = -80
L = -80 / -0.4
L = 200
Therefore, at a labor input of L = 200, the total output of cut-flower reaches its maximum.
c. To find the maximum achievable amount of cut-flower production, substitute the value of L (200) into the production function:
Q = -0.2(200)^2 - 1.2(10)^2 + 8(200)(10)
Calculating this will give you the maximum achievable amount of cut-flower production.