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.

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.

Suppose a company has just introduced a new line of ceramic insulators for which it has received patent protection, effectively granting the company monopoly status in the industry. The company's revenueandcost relations are given as:
TR = $300Q - $0.001Q2
TC = $9,000,000 + $20Q + $0.0004Q2
where TR is total revenue,

1) An imperfectly competitive firm has the following demandandcostfunctions:
P=230-20Q
C=50+30Q
a. What is optimumoutput?
b. What is equilibrium price of this output per unit?
c. What is optimal revenue?
d. What is total profit?
2) A firm in a perfectly-competitive industry where market price of output prevailing

A Monopolist's Demandand Total Cost functions are:
P= 1624 -4Q
TC= 22,000 + 24Q -4Q(squared) + 1/3Q (to the third power)
Where Q is output produced and sold
a. At what level of outputand sales (Q) andprice (P) will Total Profits be maximized?
b. At What level of outputand sales (Q) andprice (P) will Total Rev

Assume a monopolist with the following:
Qd =100-10p
TC = 1 + 2Q
Find the following:
a) Price at profit maximizing output
b) Profit maximizing output
c) Total Revenue at profit maximizing output
d) Total Cost at profit maximizing output
e)Profit

Please refer attached file for diagram.
The follwing diagram shows the cost structure of a mononpoly firm as well as market demand. Identify on the graph and calculate the following:
a. Profit-maximizing outputlevel
b. Profit-maximizing price
c. Total Revenue
d. Total Cost
e. Total profit or loss.
State the correct

As manager of Citywide Racquet Club, you must determine the best price to charge for locker rentals. Assume that the (marginal) cost of providing lockers is 0.
The monthly demand for lockers is estimated to be: Q= 100-2P where P is the monthly rental priceand Q is the number of lockers rented per month.
a. What price wou

A monopolist's demand function is given by
P = 80-3Q
(with MR = 80-6Q).
Its total cost function is
TC = 20Q + 200
(with MC = 20).
(i) Using algebra determine the profit maximizing output, priceand optimal profit for the firm.
(ii) Suppose that instead of maximizing profit, the firm wants to maximize total revenue

A firm has two plants with the following marginal costfunctions: MC1 = 20 +2Q1, MC2 = 10 +5Q2
Where MC1 is marginal cost in the first plant, MC2 is marginal cost in the second plant, Q1 is output in the first plant and Q2 is output in the second plant.
a.If the firm is minimizing its costand if it is producing 5 units of o

10. Cleaners R Us. offers professional motel room cleaning to motel owners in Danville, Illinois. The company estimates that each additional room 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 Quan