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.
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?
1 answer
1 answer