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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 \(\$2\).

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Most popular questions from this chapter

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?

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