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

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.?

#### 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.