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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Marginal Cost
In the context of perfect competition, marginal cost (MC) plays a critical role. It refers to the cost added by producing one additional unit of a product. For the firms in our videotape-copying industry, the marginal cost per tape is constant at $10. This means each additional tape copied costs the firm exactly $10.
Marginal cost is important because, in a perfectly competitive market, firms aim to produce where the price equals the marginal cost in the long-run equilibrium. This ensures that firms earn normal profits, covering both fixed and variable costs, without making excess profits.
Demand Equation
The demand equation helps us understand how much quantity is demanded at different prices. Initially, the demand for tapes is given by the equation \(Q = 1,050 - 50P\). This linear equation shows that as the price \(P\) increases, the quantity demanded \(Q\) decreases, reflecting the law of demand.
A change in the demand equation, such as \(Q = 1,600 - 50P\), indicates a shift. This can be due to factors like changes in consumer preferences. In our case, a higher intercept in the demand equation signifies increased demand, affecting the equilibrium price and quantity.
Long-Run Equilibrium
Long-run equilibrium occurs when firms in a market produce at a level where they make normal profits. In perfect competition, this happens when the price \(P\) equals both the marginal cost \(MC\) and the average total cost. This state ensures no firm has a reason to enter or exit the market.
In our scenario, we find the equilibrium by setting the price equal to \(MC + r\) (where \(r\) is the royalty fee). Solving the demand and price equations helps determine the equilibrium quantity \(Q\) and price \(P\). Initially, the equilibrium price is \\(11.20 with a quantity of 600 tapes, and after the demand increase, it's \\)12 with 1,000 tapes.
Producer Surplus
Producer surplus represents the difference between what producers are willing to accept for a good compared to what they actually receive. It's a reflection of the benefits producers gain by selling at the market price rather than the lowest price at which they would be willing to sell.
In the videotape-copying industry, an increase in producer surplus indicates that firms benefit from a higher price due to increased demand. Analyzing changes in producer surplus helps assess how much producers gain financially from shifts in market conditions. This is crucial for understanding incentives for firms and the overall health of the market.

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Most popular questions from this chapter

The domestic demand for portable radios is given by $$Q=5,000-100 \mathrm{P}$$ where price \(\mathrm{(P)}\) is measured in dollars and quantity \(\mathrm{(Q)}\) is measured in thousands of radios per year. The domestic supply curve for radios is given by $$Q=150 \mathrm{P}$$ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $$\$ 10$$ per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $$\$ 5$$ tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to "voluntarily" limit the portable radios they export to \(1,250,000\) per year? Explain how this differs from the case of a tariff.

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).

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?

demand is given by \\[ Q=1,500-50 P \\] a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price \(\left(P^{*}\right) ?\) The total industry output \(\left(Q^{*}\right) ?\) The output of each firm \(\left(q^{*}\right) ?\) The number of firms? The profits of each firm? c. The short-run total cost function associated with each firm's long-run equilibrium output is given by \\[ C(q)=0.5 q^{2}-10 q+200 \\] Calculate the short-run average and marginal cost function. At what output level does short-run average cost reach a minimum? d. Calculate the short-run supply function for each firm and the industry short-run supply function. e. Suppose now that the market demand function shifts upward to \(Q=2,000-50 P .\) Using this new demand curve, answer part (b) for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry?

Throughout this chapter's analysis of taxes we have used per-unit taxes-that is, a tax of a fixed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes-that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of \(t(t=0.05 \text { for a } 5\) percent tax), the gap between the price demanders pay and what suppliers receive is given by \(P_{D}=(1+t) P_{S}\) a. Show that for an ad valorem tax \\[ \frac{d \ln P_{D}}{d t}=\frac{e_{S}}{e_{S}-e_{D}} \quad \text { and } \quad \frac{d \ln P_{S}}{d t}=\frac{e_{D}}{e_{S}-e_{D}} \\] b. Show that the excess burden of a small tax is \\[ D W=-0.5 \frac{e_{D} e_{S}}{e_{S}-e_{D}} t^{2} P_{0} Q_{0} \\] c. Compare these results with those derived in this chapter for a unit tax.
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