Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 4

Suppose you are the manager of a watchmaking firm operating in a competitive market. Your cost of production is given by \(C=200+2 q^{2}\), where \(q\) is the level of output and \(C\) is total cost. (The marginal cost of production is \(4 q ;\) the fixed cost is \(\$ 200 .\) ) a. If the price of watches is \(\$ 100,\) how many watches should you produce to maximize profit? b. What will the profit level be? c. At what minimum price will the firm produce a positive output?

Short Answer

Expert verified
To maximize the profit, around 12 watches should be produced. The maximum profit given this output would be approximately $400. The firm would need to price the watches at a minimum of $200 to produce a positive output.

Step by step solution

01

Calculation of quantity to maximize profit

Start by setting up the profit equation, which is profit = total revenue - total cost. Since total revenue is output (q) times price (P), and total cost is given by the problem as \(C=200+2q^2\), the profit equation becomes profit = qP - (200+2q^2). Given that the price of watches is $100, substitute P=100 to find q. Taking derivative of the profit equation with respect to q, setting it to 0 and solve for q to get the quantity to maximize profit.
02

Calculation of maximum profit level

Once you have found the value for q that maximizes profit, substitute this value back into the profit equation to find the maximum profit. This is done by plugging the optimal q into the profit = qP - (200+2q^2) and evaluating.
03

Determination of minimum price for positive output

To produce a positive output, the total revenue should exceed the fixed cost of producing the good. Thus, set the total revenue (qP) equal to the fixed cost (200) and solve for the price P, assuming the quantity produced q is at least 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that a competitive firm's marginal cost of producing output \(q\) is given by \(\mathrm{MC}(q)=3+2 q\). Assume that the market price of the firm's product is \(\$ 9\) a. What level of output will the firm produce? b. What is the firm's producer surplus? c. Suppose that the average variable cost of the firm is given by \(\mathrm{AVC}(q)=3+q .\) Suppose that the firm's fixed costs are known to be \(\$ 3 .\) Will the firm be earning a positive, negative, or zero profit in the short run?

A number of stores offer film developing as a service to their customers. Suppose that each store offering this service has a cost function \(C(q)=50+0.5 q+0.08 q^{2}\) and a marginal cost \(M C=0.5+0.16 q\) a. If the going rate for developing a roll of film is \(\$ 8.50\) is the industry in long-run equilibrium? If not, find the price associated with long- run equilibrium. b. Suppose now that a new technology is developed which will reduce the cost of film developing by 25 percent. Assuming that the industry is in longrun equilibrium, how much would any one store be willing to pay to purchase this new technology?

Suppose you are given the following information about a particular industry: \\[ \begin{array}{ll} Q^{D}=6500-100 P & \text { Market demand } \\ Q^{S}=1200 P & \text { Market supply } \end{array} \\] \\[ C(q)=722+\frac{q^{2}}{200} \quad \text { Firm total cost function } \\] \\[ M C(q)=\frac{2 q}{200} \quad \text { Firm marginal cost function } \\] Assume that all firms are identical and that the market is characterized by perfect competition. a. Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of each firm. b. Would you expect to see entry into or exit from the industry in the long run? Explain. What effect will entry or exit have on market equilibrium? c. What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain. d. What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain.

Consider a city that has a number of hot dog stands operating throughout the downtown area. Suppose that each vendor has a marginal cost of \(\$ 1.50\) per hot dog sold and no fixed cost. Suppose the maximum number of hot dogs that any one vendor can sell is 100 per day. a. If the price of a hot dog is \(\$ 2,\) how many hot dogs does each vendor want to sell? b. If the industry is perfectly competitive, will the price remain at \(\$ 2\) for a hot dog? If not, what will the price be? c. If each vendor sells exactly 100 hot dogs a day and the demand for hot dogs from vendors in the city is \(Q=4400-1200 P,\) how many vendors are there? d. Suppose the city decides to regulate hot dog vendors by issuing permits. If the city issues only 20 permits and if each vendor continues to sell 100 hot dogs a day, what price will a hot dog sell for? e. Suppose the city decides to sell the permits. What is the highest price that a vendor would pay for a permit?

A sales tax of \(\$ 1\) per unit of output is placed on a particular firm whose product sells for \(\$ 5\) in a competitive industry with many firms. a. How will this tax affect the cost curves for the firm? b. What will happen to the firm's price, output, and profit? c. Will there be entry or exit in the industry?
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free