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# Cost and Revenue Functions: Calculating Optimum Output and Price Level

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Sparkling Pipes, Inc. offers professional furnace duct cleaning to home owners in Danville, Illinois. The company estimates that each additional room of ducts it cleans costs the firm \$10. The owner's daughter did a study and estimated the firm's demand could be described by the following equation, where P stands for price, and Q for Quantity demanded.

She also estimated the marginal revenue equation for the company could be described by the equation below the demand equation. Lastly, since each additional room costs \$10 to clean, she also derived a marginal cost equation.

Demand, P = \$40 - \$0.001Q (or Q = 40,000 - 1,000P)
marginal revenue, MR = \$40 - \$0.002Q
marginal cost, MC = \$10

a. Calculate the output level (Q - number of rooms of ducts cleaned) at which profits are maximized. (Hint: remember that profits are maximized at that output where marginal revenue (MR) = marginal cost (MC))

b. In order to sell the amount computed in part A. above; at what price would the company have to offer its service? (Hint: which curve (equation) gives the relationship between the price of a good or service and how much of it is demanded?).

c. What is this firm's total revenue at the optimum price/output computed in parts A. and B.?

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#### Solution Preview

a. Calculate the output level (Q - number of rooms of ducts cleaned) at which profits are maximized. (Hint: remember that profits are maximized at that output where marginal revenue (MR) = marginal cost (MC))

Put MR=MC ...

#### Solution Summary

This solution shows the steps to calculate profit maximizing output and price level for Sparkling Pipes. It also calculates the firm's total revenue at the optimum level.

\$2.19