1=Assume the short run production function of a firm is given by Q=KL-0.2K^2-0.1L^2. If L is variable input and K is fixed at 10
1.1=How many labour forces should the company hire to produce its maximum possible output size? And show if average productivity of labour force is increasing or decreasing at this point.
1.2=How many labour forces should the company hire to make the average productivity of the variable input maximum? And show if the marginal productivity of the labour force is increasing or decreasing at this point.
1.1=The company should hire 10 labour forces to produce its maximum possible output size. At this point, the average productivity of labour force is decreasing, since the marginal product of labour is negative.
1.2=The company should hire 5 labour forces to make the average productivity of the variable input maximum. At this point, the marginal productivity of the labour force is increasing, since the marginal product of labour is positive.
1.1 - To determine the number of labor forces required to maximize output, we need to find the quantity of labor (L) that corresponds to the maximum point on the production function.
By taking the derivative of the production function with respect to labor (L), we can find the critical point where the derivative equals zero.
Q = KL - 0.2K^2 - 0.1L^2
dQ/dL = K - 0.2(2L)
0 = K - 0.4L
Since K is fixed at 10, we can substitute that value into the equation:
0 = 10 - 0.4L
0.4L = 10
L = 25
Therefore, the company should hire 25 labor forces to produce its maximum output size.
To determine if the average productivity of the labor force is increasing or decreasing at this point, we can calculate the average productivity by dividing the total product (Q) by the number of labor forces (L).
Average Productivity = Q / L
Average Productivity = (10L - 0.2(10^2) - 0.1L^2) / L
Average Productivity = (250 - 20 - 0.1L^2) / L
Average Productivity = (230 - 0.1L^2) / L
By substituting the value of L into this equation, we can determine whether the average productivity is increasing or decreasing. However, the given information does not provide any values for K or L, making it impossible to determine the specific result.
1.2 - To find the number of labor forces that maximize the average productivity of the variable input, we need to equate the marginal productivity to zero.
To calculate the marginal productivity, we take the derivative of the production function with respect to labor (L):
MP = dQ/dL = K - 0.4L
Setting this equation equal to zero:
0 = K - 0.4L
Again, since K is fixed at 10, we can substitute that value into the equation:
0 = 10 - 0.4L
0.4L = 10
L = 25
Therefore, the company should hire 25 labor forces to maximize the average productivity of the variable input.
To determine if the marginal productivity of the labor force is increasing or decreasing at this point, we need to calculate the second derivative:
d^2Q/dL^2 = -0.4 < 0
Since the second derivative is negative, the marginal productivity of the labor force is decreasing.
I hope my clownish explanation brings a smile to your face!
To find the values for the number of labor forces and the corresponding average and marginal productivity, we'll need to differentiate the production function with respect to L and solve for the critical points.
1.1: Maximum possible output size
To find the number of labor forces that will produce the maximum output, we need to find the value of L that maximizes the production function.
Given: Q = KL - 0.2K^2 - 0.1L^2, where K is fixed at 10.
Step 1: Differentiate the production function with respect to L
dQ/dL = K - 0.2L
Step 2: Set the derivative equal to zero and solve for L
0 = K - 0.2L
0.2L = K
L = K/0.2
L = 10/0.2
L = 50
Therefore, the company should hire 50 labor forces to produce its maximum possible output size.
To determine if the average productivity of labor force is increasing or decreasing at this point, we need to take the second derivative.
Step 3: Differentiate the derivative with respect to L (d^2Q/dL^2)
d^2Q/dL^2 = -0.2
If d^2Q/dL^2 < 0, it means the average productivity is decreasing at this point. In our case, -0.2 < 0, so the average productivity of the labor force is decreasing when the company hires 50 labor forces.
1.2: Maximum average productivity of labor force
To find the number of labor forces that will maximize the average productivity of the variable input (L), we need to find the value of L that maximizes the derivative of the production function with respect to L.
Given: dQ/dL = K - 0.2L
Step 1: Set the derivative equal to zero and solve for L
0 = K - 0.2L
0.2L = K
L = K/0.2
L = 10/0.2
L = 50
Therefore, the company should hire 50 labor forces to make the average productivity of the variable input (L) maximum.
To determine if the marginal productivity of the labor force is increasing or decreasing at this point, we need to examine the derivative.
Step 3: Differentiate the production function with respect to L again
d^2Q/dL^2 = -0.2
If d^2Q/dL^2 < 0, it means the marginal productivity is decreasing at this point. In our case, -0.2 < 0, so the marginal productivity of labor force is decreasing when the company hires 50 labor forces.
To find the answers to these questions, we need to first differentiate the short run production function with respect to labor (L) to find the marginal product of labor (MPL) and then the average product of labor (APL).
1.1- To determine the number of labor forces that the company should hire to produce its maximum possible output size, we need to find the point where the marginal product of labor (MPL) equals zero.
First, let's find the MPL by differentiating the production function with respect to L:
Q = KL - 0.2K^2 - 0.1L^2
dQ/dL = K - 0.2L
Setting dQ/dL equal to zero and substituting K = 10 (fixed input), we can solve for L:
K - 0.2L = 0
10 - 0.2L = 0
0.2L = 10
L = 50
Therefore, the company should hire 50 labor forces to produce its maximum output size.
To determine if the average productivity of the labor force is increasing or decreasing at this point, we need to examine the average product of labor (APL). APL is given by Q/L, where Q is the output and L is the number of labor forces.
In this case, APL = Q/L = (KL - 0.2K^2 - 0.1L^2)/L
Substituting K = 10 and L = 50, we can calculate the value of APL.
1.2- To find the number of labor forces the company should hire to maximize the average productivity of the variable input (APL), we need to determine the point where the marginal product of labor (MPL) equals the average product of labor (APL).
First, let's find the MPL as we did in the previous question:
MPL = K - 0.2L
To find the number of labor forces (L) at which MPL equals APL, we set MPL equal to APL:
K - 0.2L = APL (substituting APL from question 1.2)
Now, we need to find APL by substituting K = 10 and L with an unknown variable, which we will solve for:
APL = (10L - 0.2 * 10^2 - 0.1L^2)/L
Setting MPL equal to APL:
K - 0.2L = APL
Substituting APL with the expression above, we get:
10 - 0.2L = (10L - 0.2 * 10^2 - 0.1L^2)/L
Now, we can solve this equation to find the value of L at which MPL equals APL.