George, 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
Where Q = number of car washes per hour and L = number of workers.
a) Generate a schedule showing total product, average product and marginal product. (5 marks)
b) The price of a basic car wash is $50. How many workers should he hire if each worker is
paid $60/hour? (3 marks)
c) He is considering hiring students on a part–time basis for $40/hour, do you think he
should do so? Explain. (2 marks)
a) To generate a schedule showing total product, average product, and marginal product, we can use the given production function:
Q = 6L - 0.5L^2
To calculate the total product (TP), we need to substitute different values of L into the production function and calculate Q. Let's assume we start with L = 0 and increase it by 1 each time:
L = 0: Q = 6(0) - 0.5(0)^2 = 0
L = 1: Q = 6(1) - 0.5(1)^2 = 5.5
L = 2: Q = 6(2) - 0.5(2)^2 = 10
L = 3: Q = 6(3) - 0.5(3)^2 = 12.5
L = 4: Q = 6(4) - 0.5(4)^2 = 14
So the total product schedule (TP) is as follows:
L | Q
--------------
0 | 0
1 | 5.5
2 | 10
3 | 12.5
4 | 14
To calculate the average product (AP), we divide the total product by the number of workers (L):
AP = Q / L
Using the values from the total product schedule, we can calculate the average product:
L = 0: AP = 0 / 0 (undefined)
L = 1: AP = 5.5 / 1 = 5.5
L = 2: AP = 10 / 2 = 5
L = 3: AP = 12.5 / 3 ≈ 4.17
L = 4: AP = 14 / 4 = 3.5
So the average product schedule (AP) is as follows:
L | AP
--------------
0 | undefined
1 | 5.5
2 | 5
3 | 4.17
4 | 3.5
To calculate the marginal product (MP), we find the change in total product when adding one more worker:
MP = ΔQ / ΔL
Using the values from the total product schedule, we can calculate the marginal product:
L = 0: MP = ΔQ = Q(1) - Q(0) = 5.5 - 0.5(1)^2 = 5.5
L = 1: MP = ΔQ = Q(2) - Q(1) = 10 - 5.5 = 4.5
L = 2: MP = ΔQ = Q(3) - Q(2) = 12.5 - 10 = 2.5
L = 3: MP = ΔQ = Q(4) - Q(3) = 14 - 12.5 = 1.5
So the marginal product schedule (MP) is as follows:
L | MP
--------------
0 | 5.5
1 | 4.5
2 | 2.5
3 | 1.5
b) To determine how many workers George should hire, we need to consider the marginal product of labor and compare it to the labor cost. The marginal product of labor is the additional output produced when adding one more unit of labor.
In this case, the labor cost per hour is $60, and each car wash is sold for $50. To determine the optimal number of workers, we need to find the number of workers where the marginal product of labor is equal to or greater than the ratio of the labor cost to the revenue generated per car wash:
MP ≥ (Labor Cost Per Hour) / (Price per Car Wash)
Let's calculate the marginal product for each worker:
L = 0: MP = 5.5
L = 1: MP = 4.5
L = 2: MP = 2.5
L = 3: MP = 1.5
As the number of workers increases, the marginal product decreases. To cover labor costs, we need the marginal product to be equal to or greater than the labor cost divided by the price per car wash:
MP ≥ (60) / (50) = 1.2
Therefore, George should hire 3 workers since the marginal product of the third worker (1.5) is the last value that is greater than or equal to 1.2.
c) George is considering hiring students on a part-time basis for $40 per hour. To determine whether he should hire them or not, we need to compare their cost to their contribution to the production process.
Using the same calculation as in part b), we need the marginal product to be equal to or greater than the labor cost divided by the price per car wash:
MP ≥ (Labor Cost Per Hour) / (Price per Car Wash)
Let's recalculate the marginal product for each worker:
L = 0: MP = 5.5
L = 1: MP = 4.5
L = 2: MP = 2.5
L = 3: MP = 1.5
Given that the students' cost is $40 per hour, we can compare their marginal product to the labor cost:
For L = 0: MP = 5.5 ≥ (40) / (50) = 0.8, so hiring a student may not be necessary.
For L = 1: MP = 4.5 ≥ (40) / (50) = 0.8, so hiring a student may not be necessary.
For L = 2: MP = 2.5 < (40) / (50) = 0.8, so hiring a student may be beneficial.
For L = 3: MP = 1.5 < (40) / (50) = 0.8, so hiring a student may be beneficial.
Therefore, based on the marginal product, it seems beneficial for George to hire students on a part-time basis for $40 per hour as their contribution to the production process is still greater than their cost.