Ever Kleen Pool Services provides weekly swimming pool maintenance in Atlanta. 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 four month service contract is $115.
Ever Kleen Pool Services has fixed costs of $3500. The manager if Ever Kleen has estimated the following marginal cost function for Ever Kleen, using data for the last two years:
SMC=125 - 0.42Q +0.0021Q^2 (suppose to be exponent 2)
Where SMC is measured in dollars and Q is the number of pools serviced each summer. Each of estimated coefficients is statistically significant at the 5 percent level.
a.Given the estimated marginal cost function, what is the average variable cost function for EverKleen?
b.At what output level does AVC reach its minimum value? What is the value of AVC at its minimum point?
c.Should the manager of EverKleen continue to operate, or should the firm shut down? Explain.
d.The manager of EverKleen finds two output levels that appear to be optimal. What are these levels of output and which one is actually optimal?
e.How much profit (or loss) can the manager of EverKleen Pool Services expect to earn?
f.Suppose EverKleen's fixed costs rise to $4000. How does this affect the optimal level of output ? Explain
post the answer
a. To find the average variable cost (AVC) function, we need to divide the total variable cost (TVC) by the quantity of pools serviced (Q).
Since the marginal cost (MC) function is given as SMC = 125 - 0.42Q + 0.0021Q^2, the TVC function can be found by integrating the MC function with respect to Q.
Integrating the MC function gives:
TVC = ∫(125 - 0.42Q + 0.0021Q^2)dQ
TVC = 125Q - 0.21Q^2 + 0.0007Q^3 + C
To find the constant of integration (C), we can use the information provided that EverKleen has fixed costs of $3500. When Q = 0, TVC should equal fixed costs, so:
3500 = 0 + 0 + 0 + C
C = 3500
Substituting the value of C back into the TVC function, we have:
TVC = 125Q - 0.21Q^2 + 0.0007Q^3 + 3500
To find AVC, we divide TVC by Q:
AVC = TVC / Q
AVC = (125Q - 0.21Q^2 + 0.0007Q^3 + 3500) / Q
AVC = 125 - 0.21Q + 0.0007Q^2 + 3500/Q
b. To determine the output level at which AVC reaches its minimum value, we need to find the derivative of the AVC function with respect to Q and set it equal to zero.
Taking the derivative of AVC with respect to Q, we get:
dAVC/dQ = -0.21 + (0.0014Q - 3500/Q^2)
Setting this derivative equal to zero and solving for Q, we have:
-0.21 + (0.0014Q - 3500/Q^2) = 0
0.0014Q - 3500/Q^2 = 0.21
Simplifying and rearranging, we get:
0.0014Q^3 - 3500 = 0.21Q^2
This is a nonlinear equation, and we need to use numerical methods or a graphing calculator to find the solution. Solving this equation will give us the output level at which AVC reaches its minimum value.
c. To determine if EverKleen should continue to operate or shut down, we need to consider whether the firm can cover its total costs (fixed costs + variable costs) with the revenue it generates.
The firm's total costs (TC) can be calculated as:
TC = Fixed Costs + TVC
TC = $3500 + (125Q - 0.21Q^2 + 0.0007Q^3)
The revenue the firm generates can be calculated as:
Revenue = Price per contract * Quantity (Q)
Given that the market price for the four-month service contract is $115, the revenue function can be written as:
Revenue = $115Q
To determine whether the firm should continue to operate or shut down, we compare TC and Revenue.
If TC is less than Revenue, the firm should continue to operate. If TC is greater than Revenue, the firm should shut down.
d. If the manager finds two output levels that appear to be optimal, it means that the marginal cost curve intersects the average total cost (ATC) curve at two points. These points indicate the output levels at which the firm can achieve the lowest average total cost.
To determine which one of the two output levels is actually optimal, we need to compare the corresponding ATC values. The output level with the lower ATC represents the optimal level of output.
e. To calculate the profit or loss that the manager of EverKleen Pool Services can expect to earn, we need to subtract the total cost (TC) from the total revenue (TR).
Profit = Total Revenue - Total Cost
Profit = (Price per contract * Quantity) - (Fixed Costs + TVC)
Profit = ($115Q) - ($3500 + (125Q - 0.21Q^2 + 0.0007Q^3))
Substituting the values and solving accordingly will give the profit (or loss) that the manager can expect to earn.
f. If EverKleen's fixed costs rise to $4000, this will affect the optimal level of output. As fixed costs increase, the minimum ATC level will also increase. This means that the firm needs to produce at a higher output level to minimize its average total cost.
The increase in fixed costs will shift the ATC curve upwards, leading to a higher optimal level of output. The manager needs to assess the impact on the new ATC curve and adjust the output level accordingly to minimize costs and maximize profitability.