Everkleen Pool Services (EPS) provides weekly swimming pool maintenance in Jeddah. Dozens of firms provide this service. The service is standardized; each company cleans the pool and maintains the proper levels of chemicals in the water. The service is typically sold as a four-month summer contract. The market price for the 4-month service contract is US$115.
EPS has fixed costs of US$3500. The manager has estimated the following marginal cost function for EPS, using data for the last two years:
MC = 125 – 0.42Q + 0.0021Q2
Where MC is measured in dollars and Q is the number of pools serviced each summer. Each of the estimated coefficients is statistically significant at the 95 percent confidence level.
• Given the estimated marginal cost function, what is the average variable cost function for EPS?
• At what output level does AVC reach its minimum value? What is the value of AVC at its minimum point?
• Should the manager of EPS continue to operate, or should the firm shut down? Explain.
• The manager of EPS finds two output levels that appear to be optimal. What are these levels of output and which one is actually optimal?
• How much profit (or loss) can the manager of EPS expect to earn?
• Suppose that EPS fixed costs rise to US$4000. How does this affect the optimal level of output? Explain.
To find the average variable cost (AVC) function, we need to divide the marginal cost (MC) function by the quantity (Q).
AVC = MC / Q
Using the given MC function:
MC = 125 – 0.42Q + 0.0021Q^2
Dividing both sides by Q:
AVC = (125Q – 0.42Q^2 + 0.0021Q^3) / Q
Simplifying:
AVC = 125 – 0.42Q + 0.0021Q^2
To find the output level at which AVC reaches its minimum value, we need to take the derivative of the AVC function with respect to Q and set it equal to zero.
dAVC/dQ = -0.42 + 0.0042Q = 0
Solving for Q:
0.0042Q = 0.42
Q = 100
So, AVC reaches its minimum value at an output level of 100 pools serviced. To find the value of AVC at its minimum point, substitute Q = 100 into the AVC function:
AVC = 125 – 0.42(100) + 0.0021(100)^2
AVC = 125 – 42 + 0.21(100)
AVC = 83
Therefore, AVC reaches its minimum value of 83 at an output level of 100 pools serviced.
To determine whether the manager of EPS should continue to operate or shut down, we need to compare the total revenue (TR) with the total cost (TC). If TR is greater than TC, the manager should continue to operate; otherwise, the firm should shut down.
Total cost (TC) can be calculated by adding fixed costs (FC) to the variable costs (VC), where VC is equal to the average variable cost (AVC) multiplied by the quantity (Q).
TC = FC + (AVC * Q)
FC = $3500 (given)
Suppose the summer contract price is $115, and the firm services Q pools each summer. The total revenue (TR) can be calculated as follows:
TR = Price * Quantity
TR = $115 * Q
Comparing TR and TC will allow us to determine whether the firm should continue operating or shut down.
If the manager has found two output levels that appear to be optimal, we need more information about how these levels were determined in order to identify the actual optimal level.
To calculate the profit or loss, we need to subtract the total cost (TC) from the total revenue (TR):
Profit = TR - TC
If Profit is positive, then the manager can expect to earn a profit. If Profit is negative, then the manager can expect to incur a loss.
If the fixed costs of EPS rise to $4000, it will not directly affect the optimal level of output determined by the AVC function. However, the increase in fixed costs will add to the overall total costs (TC), potentially affecting the profitability of the firm. The manager should recalculate the new TC by adding the increased fixed costs to the variable costs based on the new AVC function, and then compare the total cost with the total revenue to make a decision on whether to continue operating or shut down.