Suppose the production function for widgets is given by:
Q = KL - 0.8K^2 - 0.2L^2
where Q represents the quantity of widgets produced, K represents the annual capital input and L represents annual labor input.
(a) For K = 10, what is the average productivity of labor equal to?
-do i substitute k=10 into Q, and then what do i do to find the average productivity of labor?
(b) At what level of labor input does this average productivity reach a maximum? How many widgets are produced at this point?
(c) Again assuming that K = 10, what is the marginal product of labor equal to?
A.Q=40 &L=20
B.Q=45 &L=25
Output is maximum when L is 25, and the amount of output produced from this labor is 45 units.
(a) To find the average productivity of labor, you need to substitute K = 10 into the production function Q = KL - 0.8K^2 - 0.2L^2.
Substituting K = 10 into the production function, we get:
Q = 10L - 0.8(10^2) - 0.2L^2
Simplifying, we have:
Q = 10L - 80 - 0.2L^2
The average productivity of labor is equal to the quantity of widgets produced (Q) divided by the amount of labor input (L):
Average Productivity of Labor = Q / L
(b) To find the level of labor input at which the average productivity reaches a maximum, we need to differentiate the production function with respect to L and set it equal to zero, and then solve for L.
Differentiating Q = 10L - 80 - 0.2L^2 with respect to L, we get:
dQ/dL = 10 - 0.4L
Setting dQ/dL equal to zero and solving for L:
10 - 0.4L = 0
0.4L = 10
L = 25
Therefore, the level of labor input at which the average productivity reaches a maximum is L = 25.
To find the quantity of widgets produced at this point, substitute L = 25 into the production function:
Q = 10(25) - 80 - 0.2(25)^2
Q = 250 - 80 - 125
Q = 45
At L = 25, 45 widgets are produced.
(c) To find the marginal product of labor, you need to differentiate the production function with respect to L.
Differentiating Q = 10L - 80 - 0.2L^2 with respect to L, we get:
dQ/dL = 10 - 0.4L
Therefore, the marginal product of labor is given by the derivative of the production function with respect to L, which is 10 - 0.4L.
To find the average productivity of labor, you need to divide the total output (Q) by the amount of labor input (L). To do this, follow these steps:
(a)
1. Substitute K = 10 into the given production function Q = KL - 0.8K^2 - 0.2L^2.
2. This will give you an expression solely in terms of L: Q = 10L - 0.8(10)^2 - 0.2L^2.
3. Simplify the expression if necessary.
4. Divide the quantity produced (Q) by the labor input (L). The result is the average productivity of labor.
(b)
To find the level of labor input at which average productivity reaches a maximum and the corresponding quantity produced, you need to take the derivative of the production function with respect to labor (L) and find the maximum point. Follow these steps:
1. Differentiate the production function with respect to L.
2. Set the derivative equal to zero and solve for L (the labor input).
3. Plug in the value of L into the production function to find the quantity of widgets produced (Q) at this point.
(c)
To find the marginal product of labor (MPL) when K = 10, you need to take the derivative of the production function with respect to labor (L) and substitute K = 10. Follow these steps:
1. Differentiate the production function with respect to L, considering K as a constant.
2. Substitute K = 10 into the derivative expression for MPL.
3. Simplify the expression if necessary.
4. The resulting expression is the marginal product of labor (MPL) when K = 10.