Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 12: Problem 7

The perfectly competitive videotape-copying industry is composed of many firms that can copy five tapes per day at an average cost of \(\$ 10\) per tape. Each firm must also pay a royalty to film studios, and the per-film royalty rate \((r)\) is an increasing function of total industry output (Q): \\[ r=0.002 Q \\] Demand is given by \\[ Q=1,050-50 P \\] a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape firms will there be? What will the per-film royalty rate be? b. Suppose that demand for copied tapes increases to \\[ Q=1,600-50 P \\] In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape firms are there? What is the per-film royalty rate? c. Graph these long-run equilibria in the tape market, and calculate the increase in producer surplus between the situations described in parts (a) and (b). d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as \(Q\) expands incrementally from its level in part (b) to its level in part (c). e. Suppose that the government institutes a \(\$ 5.50\) per-film tax on the film-copying industry. Assuming that the demand for copied films is that given in part (a), how will this tax affect the market equilibrium? f. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? g. Show that the loss of producer surplus as a result of this tax is borne completely by the film studios. Explain your result intuitively.

Short Answer

Expert verified
Answer: In the initial long-run equilibrium, the equilibrium price is $11.20 per tape, the per-film royalty rate is $1.20, and there are 120 tape-copying firms in the industry.

Step by step solution

01

Interpret the given information

The industry has many firms with the same cost structure. The cost of copying a tape is constant (\(\$10\) per tape) plus a royalty fee \((r)\) dependent on the total industry output \((Q)\). The demand equation is given by \(Q=1,050-50P\).
02

Write the marginal cost (MC) and royalty fee function

Since each firm can copy five tapes per day at an average cost of \(\$10\), the marginal cost (MC) of their production is also \(\$10\). The royalty fee is given by the increasing function \(r=0.002Q\).
03

Find the equilibrium price and quantity

In a perfectly competitive market, firms produce at their marginal cost in the long-run equilibrium. So, we need to find the \(P\) and \(Q\) when \(MC=10+r\), which equates to \(P=10 + 0.002Q\). Substitute this price (\(P\)) equation into the demand equation: \(Q = 1,050 - 50(10 + 0.002Q)\) Solve for \(Q\): \(Q = 600\) Now, substitute the value of \(Q\) into the price equation to find \(P\): \(P = 10 + 0.002(600)\) \(P = 11.20\)
04

Calculate the per-film royalty rate

Use the royalty fee equation: \(r = 0.002(600)\) \(r = 1.20\)
05

Calculate the number of firms

Since each firm can copy five tapes per day, the number of firms \((n)\) required to reach total industry output \((Q)\) is: \(n = \frac{Q}{5} = \frac{600}{5} = 120\) Thus, in the long-run equilibrium there are 120 tape-copying firms, the equilibrium price is \(\$11.20\) per tape, and the per-film royalty rate is \(\$1.20\). #Part b - New long-run equilibrium with increased demand#
06

Rewrite the new demand equation

The new demand equation is given by: \(Q=1,600-50P\)
07

Calculate the new equilibrium price and quantity

We still have the same price equation (\(P=10 + 0.002Q\)). Substitute the new demand equation into the price equation: \(Q = 1,600 - 50(10 + 0.002Q)\) Solve for \(Q\): \(Q = 1,000\) Now, substitute the value of \(Q\) into the price equation to find \(P\): \(P = 10 + 0.002(1,000)\) \(P = 12\)
08

Calculate the new per-film royalty rate

Use the royalty fee equation with the new \(Q\) value: \(r = 0.002(1,000) = 2\) The new per-film royalty rate is \(\$2\).
09

Calculate the number of new firms

As before, each firm can copy five tapes per day. Therefore, the required number of firms is: \(n = \frac{Q}{5} = \frac{1,000}{5} = 200\) Thus, in the new long-run equilibrium there are 200 tape-copying firms, the equilibrium price is \(\$12\) per tape, and the per-film royalty rate is \(\$2\).

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 there are 1,000 identical firms producing diamonds. Let the total cost function for each firm be given by \\[ C(q, w)=q^{2}+w q \\] where \(q\) is the firm's output level and \(w\) is the wage rate of diamond cutters. a. If \(w=10\), what will be the firm's (short-run) supply curve? What is the industry's supply curve? How many diamonds will be produced at a price of 20 each? How many more diamonds would be produced at a price of \(21 ?\) b. Suppose the wages of diamond cutters depend on the total quantity of diamonds produced, and suppose the form of this relationship is given by \\[ w=0.002 Q \\] here \(Q\) represents total industry output, which is 1,000 times the output of the typical firm. In this situation, show that the firm's marginal cost (and short-run supply) curve depends on \(Q\). What is the industry supply curve? How much will be produced at a price of \(20 ?\) How much more will be produced at a price of \(21 ?\) What do you conclude about the shape of the short-run supply curve?

A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by \\[ Q=160,000-10,000 P \\] a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one firm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each firm has a supply curve that shows the quantity the firm will supply \(\left(q_{i}\right)\) as a function of market price. The specific form of this supply curve is given by \\[ q_{i}=-200+50 P \\] Using this short-run supply response, supply revised answers to (a)-(d).

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form \\[ C(q)=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10 \\] a. Calculate the firm's short-run supply curve with \(q\) as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by \(Q=-200 P+8,000 .\) What will be the short-run equilibrium price-quantity combination?

The handmade snuffbox industry is composed of 100 identical firms, each having short-run total costs given by \\[ S T C=0.5 q^{2}+10 q+5 \\] and short-run marginal costs given by \\[ S M C=q+10 \\] where \(q\) is the output of snuffboxes per day. a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by \\[ Q=1,100-50 P \\] What will be the equilibrium in this marketplace? What will each firm's total short-run profits be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry profits plus industry short-run fixed costs. e. Suppose the government imposed a \(\$ 3\) tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run profits in the snuffbox industry. Why do fixed costs not enter into this computation of the change in short-run producer surplus?

One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into producer's supply response. To examine this possibility, assume that quantity demanded in period \(t\) depends on price in that period \(\left(Q_{t}^{D}=a-b P_{t}\right)\) but that quantity supplied depends on the previous period's price-perhaps because farmers refer to that price in planting a crop \(\left(Q_{t}^{S}=c+d P_{t-1}\right)\) a. What is the equilibrium price in this model \(\left(P^{*}=P_{t}=P_{t-1}\right)\) for all periods, \(t\) b. If \(P_{0}\) represents an initial price for this good to which suppliers respond, what will the value of \(P_{1}\) be? c. By repeated substitution, develop a formula for any arbitrary \(P_{t}\) as a function of \(P_{0}\) and \(t\) d. Use your results from part (a) to restate the value of \(P_{t}\) as a function of \(P_{0}, P^{*},\) and \(t\) e. Under what conditions will \(P_{t}\) converge to \(P^{*}\) as \(t \rightarrow \infty ?\) f. Graph your results for the case \(a=4, b=2, c=1, d=1,\) and \(P_{0}=0 .\) Use your graph to discuss the origin of the term cobweb model.
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