What you are solving for in these problems is the optimal input combination based either on what I call Solution #1 and Solution #2; Solution #1 guarantees output maximization and Solution #2 guarantees profit maximization. Use them where appropriate. Use the optimization procedure to get answers not an Excel result.
1. The owner of the Hughes Car Wash believes that the relationship between the number of cars washed and the labor input is:
Q= -.8 + 4.5L - .3L2 where Q is the number of cars washed per hour and L is the number of people hours employed per hour.
The firm receives $5 for each car washed and the hourly wage rate for each person employed is $4.50. The cost of other inputs such as water is trivial and hence they are ignored.
a) How many people should be employed to maximize profit?
b) What will be the firm's hourly profit?
2. Find the optimal input combination for the following:
Q= 30X + 2XY -.5X2 -.5Y2 where Q is total output, X is worker hours, and Y is machine tool hours
Given that the price of X is $25.00 per unit of X and the price of Y is $40.00 per unit of Y and MR is $.50, find the optimal input combination for this production relationship.
3. In the Elwyn Company, the relationship between output (Q) and the number of hours of skilled labor (S) and unskilled labor (U) is as follows:
Q= 300S + 200U -.2S2 -.3U2
The hourly wage of skilled labor is $10.00 per unit and the hourly wage of unskilled labor is $5.00 per unit. The firm can hire as much labor as it wants at these wage rates.
A) Elwyn's chief engineer recommends that the firm hire 400 hours of skilled labor and 100 hours of unskilled labor. Evaluate this recommendation.
B) If the Elwyn Company decided to spend a total of $5,000 on skilled and unskilled labor, how many hours of each type of labor should it hire?
C) If the price of a unit of output is $10.00, how many hours of unskilled and how many hours of skilled labor should the company hire?
See attached file for full problem description.
Economics Output Maximization and Profit Maximization are explored.