Consider the following short run production function : Q=6L2-0.4L3
a. Find the value of L that maximizes out put
b. Find the value of L that maximizes marginal product
c. Find the value of L that maximizes average product
a. To find the value of L that maximizes output, we need to take the derivative of the production function with respect to L and set it equal to zero, and then solve for L.
Q = 6L^2 - 0.4L^3
Taking the derivative with respect to L:
dQ/dL = 12L - 1.2L^2
Setting it equal to zero:
12L - 1.2L^2 = 0
Dividing both sides by L:
12 - 1.2L = 0
1.2L = 12
L = 10
Therefore, the value of L that maximizes output is 10.
b. Marginal product is the derivative of the production function with respect to L:
MP = dQ/dL = 12L - 1.2L^2
To find the value of L that maximizes marginal product, we need to set the derivative equal to zero and solve for L:
12L - 1.2L^2 = 0
Dividing both sides by L:
12 - 1.2L = 0
1.2L = 12
L = 10
Therefore, the value of L that maximizes marginal product is 10.
c. Average product is given by AP = Q/L. To find the value of L that maximizes average product, we need to find the value of L that maximizes Q/L.
AP = Q/L = (6L^2 - 0.4L^3)/L = 6L - 0.4L^2
To maximize AP, we need to take the derivative of AP with respect to L and set it equal to zero:
d(AP)/dL = 6 - 0.8L = 0
0.8L = 6
L = 7.5
Therefore, the value of L that maximizes average product is 7.5.
To find the value of L that maximizes output, we need to find the value of L where the derivative of the production function with respect to L equals zero. Let's first calculate the derivative:
dQ/dL = 12L - 1.2L^2
a. Find the value of L that maximizes output:
To find the maximum of the production function, we set the derivative equal to zero and solve for L:
12L - 1.2L^2 = 0
Factor out L:
L(12 - 1.2L) = 0
Set each factor equal to zero:
L = 0 or 12 - 1.2L = 0
If L = 0, it means there is no input, which is not practical in this scenario. Therefore, we focus on the second equation:
12 - 1.2L = 0
Add 1.2L to both sides:
1.2L = 12
Divide both sides by 1.2:
L = 12 / 1.2
L = 10
Therefore, the value of L that maximizes output is 10.
b. Find the value of L that maximizes marginal product:
The marginal product is the derivative of the production function. So, we need to find the derivative of the production function again and set it equal to zero:
d²Q/dL² = 12 - 2.4L
Setting this equal to zero:
12 - 2.4L = 0
Subtract 12 from both sides:
-2.4L = -12
Divide both sides by -2.4:
L = -12 / -2.4
L = 5
Therefore, the value of L that maximizes marginal product is 5.
c. Find the value of L that maximizes average product:
The average product is output divided by input. So, we need to find the value of L that gives the maximum average product. The average product can be calculated by dividing the production function by L:
AP = Q / L
AP = (6L^2 - 0.4L^3) / L
AP = 6L - 0.4L^2
To find the maximum of the average product, we set the derivative of the average product function equal to zero:
d(AP)/dL = 6 - 0.8L = 0
6 = 0.8L
L = 6 / 0.8
L = 7.5
Therefore, the value of L that maximizes average product is 7.5.
To find the value of L that maximizes output (Q), we need to find the point at which the derivative of the production function with respect to L is equal to zero.
a. Maximize Output:
1. Take the derivative of Q with respect to L: dQ/dL = 12L - 1.2L^2
2. Set the derivative equal to zero and solve for L: 12L - 1.2L^2 = 0
3. Factor out L: L(12 - 1.2L) = 0
4. Set each factor equal to zero and solve for L:
- L = 0 (ignoring)
- 12 - 1.2L = 0
5. Solve for L: 1.2L = 12
L = 10
Thus, the value of L that maximizes the output is L = 10.
b. Maximize Marginal Product:
The marginal product is the derivative of the production function with respect to L. So we need to find the L that makes the derivative equal to zero.
1. Take the derivative of Q with respect to L: dQ/dL = 12L - 1.2L^2
2. Set the derivative equal to zero and solve for L: 12L - 1.2L^2 = 0
3. Factor out L: L(12 - 1.2L) = 0
4. Set each factor equal to zero and solve for L:
- L = 0 (ignoring)
- 12 - 1.2L = 0
5. Solve for L: 1.2L = 12
L = 10
Thus, the value of L that maximizes the marginal product is L = 10.
c. Maximize Average Product:
The average product is given by the ratio of total output (Q) to the number of units of input (L). To maximize average product, we need to find the L value at which average product is at its highest point.
1. Average Product (AP) = Q / L = (6L^2 - 0.4L^3) / L = 6L - 0.4L^2
2. Take the derivative of AP with respect to L: d(AP)/dL = 6 - 0.8L
3. Set the derivative equal to zero and solve for L: 6 - 0.8L = 0
4. Solve for L: 0.8L = 6
L = 7.5
Thus, the value of L that maximizes the average product is L = 7.5.