Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 8

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 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 units of output?

Short Answer

Expert verified
MC: \(3q^{2}-16q+30\), AC: \(q^{2}-8q+30+ \frac{5}{q}\), AVC: \(q^{2}-8q+30\). Zero output is supplied at price ranges below 14. The supply curve is represented by MC above its intersection with AVC. The firm supplies exactly 6 units of output when the price is 86.

Step by step solution

01

- Find MC, AC, and AVC

First, find the derivatives of the cost function to derive MC, AC and AVC. MC is the derivative of the total cost function. Therefore \(\mathrm{MC} = \frac{dC}{dq} = 3q^{2}-16q+30\).AC is total cost divided by quantity. Therefore, \(\mathrm{AC} = \frac{C}{q} = q^{2}-8q+30+ \frac{5}{q}\).AVC is total variable cost divided by quantity. The variable cost here is \(q^{3}-8q^{2}+30q\). So, \(AVC = \frac{q^{3}-8q^{2}+30q}{q} = q^{2}-8q+30\)
02

- Graph MC, AC and AVC

For graphing MC, AC and AVC, one can make use of softwares like Desmos or a graphing calculator. Input the equations for MC, AC and AVC to produce the graph, remembering to specify a reasonable domain. Pay attention to the intercepts and intersections between these curves.
03

- Identify Zero Output Range

The firm will supply zero output when the price is less than the minimum point on the AVC curve. Solve for q when \(AVC = q^{2}-8q+30\) is at its minimum. This is achieved by taking the derivative of AVC and solving for q equal to zero. Here, the minimum AVC is obtained at \(q = 4\), and substituting back to AVC, we find that when AVC = 14, the firm will supply zero output. Thus, at prices less than 14, the firm won't supply any output.
04

- Supply Curve Identification

The firm's supply curve can be identified in the graph from Step 2. It occurs where the price curve (a horizontal line at any given price level) intersects with the MC curve and is above AVC. Remember that a firm will only produce if the price is more than AVC.
05

- Find Specific Output Level

To find the price at which the firm would supply exactly 6 units of output, substitute \(q = 6\) in the MC function. This is because MC also represents the supply curve for perfect competition in the short run above the shutdown point. Therefore, \(P = 3*6^{2}-16*6+30 = 86\)

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 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} \\] \(C(q)=722+\frac{q^{2}}{200} \quad\) 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. 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.

A number of stores offer film developing as a service to their customers. Suppose that each store offering this service has a cost function \(C(q)=50+0.5 q+0.08 \eta^{2}\) and a marginal cost \(M C=0.5+0.16 \eta\) 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 long run equilibrium, how much would any one store be willing to pay to purchase this new technology?

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 firm produces a product in a competitive industry and has a total cost function \(C=50+4 q+2 q^{2}\) and a marginal cost function \(\mathrm{MC}=4+4 q\). At the given market price of \(\$ 20,\) the firm is producing 5 units of output. Is the firm maximizing its profit? What quantity of output should the firm produce in the long run?

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