Economics Production Schedule
The owner of a car wash is trying to decide on the number of people to employ based on the following short-run production function: Q = 6L - 0.5L2, with the corresponding marginal product of labor equation, MPL = 6 - L.
a. Generate a schedule showing total product, average product of labor, and marginal product of labor using a range of labor inputs 0 to 6.
b. Plot the production function and the APL and MPL on two graphs.
c. Suppose the price of a basic car wash is $5. How many people should be hired if each worker is paid $6 per hour?
d. If the owner can hire students on a part-time basis for $4 per hour, should he hire assuming he/she keeps the other workers and continues to pay them $5 per hour?
Take a shot, what do you think.
Hint: the value of the marginal product (VMP) is MPL*P where, in this example, P is the price of a car wash. Maximize by setting VMP = wage rate.
I want answer to this question pl.
a. To generate the schedule, we will calculate the total product (Q), average product of labor (APL), and marginal product of labor (MPL) for a range of labor inputs from 0 to 6.
Labor input (L) | Total Product (Q) | APL | MPL
0 | 0 | - | -
1 | 5 | 5 | 6-1=5
2 | 8 | 4 | 6-2=4
3 | 9 | 3 | 6-3=3
4 | 8 | 2 | 6-4=2
5 | 5 | 1 | 6-5=1
6 | 0 | 0 | 6-6=0
b. Now, let's plot the production function, APL, and MPL on two separate graphs.
Graph 1: Production Function
- The x-axis represents the labor input (L).
- The y-axis represents the total product (Q).
- Plot the points from the schedule (Labor input vs. Total Product).
- Connect the points to form a smooth curve.
Graph 2: APL and MPL
- The x-axis represents the labor input (L).
- The y-axis represents average product of labor (APL) and marginal product of labor (MPL).
- Plot the points from the schedule (Labor input vs. APL and Labor input vs. MPL) on separate lines.
- Connect the points to form smooth curves for APL and MPL respectively.
c. To determine how many people should be hired when each worker is paid $6 per hour, we need to find the level of labor where the marginal cost (MC) equals the wage rate.
MC = Wage rate
MPL = 6 - L
Let's substitute the wage rate as $6 and solve for L:
6 - L = 6
-L = 0
L = 0
Therefore, no workers should be hired if each worker is paid $6 per hour, as the marginal cost exceeds the wage rate for all levels of labor input.
d. To determine if the owner should hire part-time students at a lower wage rate while keeping the other workers and paying them $5 per hour, we need to compare the marginal product of the part-time students (MPL_student) with their wage rate.
MPL_student = 6 - L
Let's substitute the wage rate as $4 and solve for L:
6 - L = 4
-L = -2
L = 2
Therefore, if the owner can hire part-time students at $4 per hour, he/she should hire 2 students. This is because their marginal product exceeds their wage rate, indicating that their hiring would contribute positively to the owner's profit.
a. To generate the schedule showing total product, average product of labor, and marginal product of labor, we can substitute different values of labor inputs (L) into the given production function and marginal product of labor equation.
For example, let's substitute values from 0 to 6 for L:
L = 0:
Q = 6(0) - 0.5(0^2) = 0
MPL = 6 - 0 = 6
APL = 0/0 = undefined (as dividing by 0 is undefined)
L = 1:
Q = 6(1) - 0.5(1^2) = 5.5
MPL = 6 - 1 = 5
APL = 5.5/1 = 5.5
Similarly, we can calculate for other values of L:
L = 2:
Q = 6(2) - 0.5(2^2) = 8
MPL = 6 - 2 = 4
APL = 8/2 = 4
L = 3:
Q = 6(3) - 0.5(3^2) = 8.5
MPL = 6 - 3 = 3
APL = 8.5/3 ≈ 2.83
L = 4:
Q = 6(4) - 0.5(4^2) = 8
MPL = 6 - 4 = 2
APL = 8/4 = 2
L = 5:
Q = 6(5) - 0.5(5^2) = 7.5
MPL = 6 - 5 = 1
APL = 7.5/5 = 1.5
L = 6:
Q = 6(6) - 0.5(6^2) = 6
MPL = 6 - 6 = 0
APL = 6/6 = 1
b. To plot the production function, we plot the values of labor input (L) on the x-axis and the corresponding output (Q) on the y-axis. Similarly, we plot the values of APL and MPL against L.
c. To determine how many people should be hired if each worker is paid $6 per hour and the price of a basic car wash is $5, we need to compare the marginal product of labor to the wage rate.
From the given marginal product of labor equation, MPL = 6 - L, we can set MPL equal to the wage rate (in this case, $6) and solve for L:
6 - L = 6
L = 0
Since the marginal product of labor becomes zero when L = 6, it implies that beyond 6 workers, hiring additional workers will decrease the output. Therefore, the owner should hire 6 workers if each is paid $6 per hour.
d. If the owner can hire students on a part-time basis for $4 per hour, we need to compare the marginal product of labor to the wage rate of students.
From the given marginal product of labor equation, MPL = 6 - L, we can set MPL equal to the wage rate of students (in this case, $4) and solve for L:
6 - L = 4
L = 2
This implies that the owner should hire 2 full-time workers, paying them $6 per hour, and hire additional part-time student workers if needed, paying them $4 per hour.