n auto-service establishment has estimated its monthly cost function as follows:
TC = 6000 + 10 Q
where Q is the number of cars it services each months and TC represents its total cost. The firm is targeting 35,000 net monthly profit servicing 2000 cars.
a. What price should the firm charge to realize the targeted profit?
b. What would be its (cost-based) markup ratio?
b. Now suppose the demand curve the firm faces is:Q = 3000 - 50 P. Is the firm going to achieve its profit goal? Explain.
c. If your to answer to (b) is "no", what would be the optimal markup ratio for this firm?
a. To find the price the firm should charge to realize the targeted profit, we need to calculate the firm's total revenue (TR). Total revenue is equal to the price (P) multiplied by the quantity (Q). Since we know the quantity (Q) is 2000 cars, we can substitute this value into the demand curve to find the corresponding price (P):
Q = 3000 - 50P
2000 = 3000 - 50P
Solving for P, we get:
50P = 3000 - 2000
50P = 1000
P = 20
Therefore, the firm should charge a price of $20 per car to realize the targeted profit.
b. The cost-based markup ratio measures the ratio of markup over the cost. To calculate the markup, we need to subtract the cost from the price:
Markup = Price - Cost
Markup = P - TC
Using the cost function TC = 6000 + 10Q and substituting the given quantity Q = 2000, we can find the corresponding cost:
TC = 6000 + 10Q
TC = 6000 + 10(2000)
TC = 6000 + 20000
TC = 26000
Now we can calculate the markup:
Markup = P - TC
Markup = 20 - 26000
Markup = -25980
The markup ratio can be calculated by dividing the markup by the cost:
Markup ratio = Markup / Cost
Markup ratio = -25980 / 26000
Markup ratio = -0.999
Therefore, the cost-based markup ratio is approximately -0.999, indicating a negative markup which is not feasible. It suggests that the firm is operating at a loss.
c. The given demand curve is Q = 3000 - 50P. To determine if the firm will achieve its profit goal, we can substitute the price of $20 into the demand curve to calculate the corresponding quantity:
Q = 3000 - 50P
Q = 3000 - 50(20)
Q = 3000 - 1000
Q = 2000
This result indicates that the firm is able to service 2000 cars, which matches the given target. Therefore, the firm should be able to achieve its profit goal.
c. In this case, the optimal markup ratio would be zero, as the firm is already achieving its profit goal by servicing the target quantity of cars.