Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 11

Suppose that a competitive firm has a total cost function \(C(q)=450+15 q+2 q^{2}\) and a marginal cost function \(M C(q)=15+4 q .\) If the market price is \(P=\$ 115\) per unit, find the level of output produced by the firm. Find the level of profit and the level of producer surplus.

Short Answer

Expert verified
The firm will produce 25 units. The detailed calculations for profit and producer surplus require computation by plugging in the values in the appropriate formulas as shown in the solution steps.

Step by step solution

01

Output Determination

Solve for output level (q) by setting the marginal cost function equal to the market price. \(MC(q) = P\), which gives us \(15 + 4q = 115\). This can be simplified to find q.
02

Solve for Output Level

To find the output level (q), subtract 15 from both sides, which gives \(4q = 100\). Then divide by 4, which gives us \(q = 25\) units.
03

Profit Calculation

We calculate profit by subtracting total cost from total revenue. The total cost \(C(q)\) at 25 units is \(450 + 15*25 + 2*25^2\). Total revenue is the price times quantity which is \(115*25\). Subtracting total cost from total revenue gives us the profit.
04

Calculate Producer Surplus

The producer surplus is given by the area between the price and marginal cost curve from 0 to the quantity produced. In this case, the producer surplus is the integral from 0 to 25 of \(P - MC(q)\) dq. We compute this integral and get the producer surplus.

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

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?

A competitive firm has the following short-run cost function: \(C(q)=q^{3}-8 q^{2}+30 q+5\) a. Find \(\mathrm{MC}, \mathrm{AC}\), and \(\mathrm{AVC}\) and sketch them on a graph. b. At what range of prices will the firm supply zero output? c. Identify the firm's supply curve on your graph. d. At what price would the firm supply exactly 6

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?

Suppose the same firm's cost function is \(C(q)=4 q^{2}+16\) a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. (Hint: Marginal cost is given by \(\mathrm{MC}=8 q\).) b. Show the average cost, marginal cost, and average variable cost curves on a graph. c. Find the output that minimizes average cost. d. At what range of prices will the firm produce a positive output? e. At what range of prices will the firm earn a negative profit? f. At what range of prices will the firm earn a positive profit?

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?
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