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Chapter 15: Problem 3

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 \((Q)\) given by $$\mathrm{r}=.002 Q$$ Demand is given by $$d=1,050-50 \mathrm{P}$$ 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 roy alty rate be? b. Suppose demand for copied tapes increases to $$Q=1,600-50 P$$ Now, 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 pro ducer 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).

Short Answer

Expert verified
In this exercise, we analyzed a perfectly competitive industry with constant production costs and a royalty cost that increases with the total output. The goal was to find the equilibrium price and quantity, the number of tape firms, and the per-film royalty rate under two different demand scenarios. We then compared these scenarios graphically and computed the change in producer surplus. Finally, we verified that the change in producer surplus equals the increase in royalties paid. In the initial long-run equilibrium, the price of copied tapes was \(\$11.2\), the quantity was 600, there were 120 tape firms, and the per-film royalty rate was \(\$1.2\). When demand increased, the new long-run equilibrium price was \(\$12\), the quantity was 1000, there were 200 tape firms, and the per-film royalty rate was \(\$2\). After graphing the long-run equilibria and calculating the increase in producer surplus, we verified that the increase in producer surplus equaled the increase in royalties paid as the output expanded incrementally from its initial level to the level under increased demand.

Step by step solution

01

Write the equations for the supply and demand curves

The demand curve is given by: $$Q_d = 1050 - 50P$$ The cost for each firm to produce a tape is \(\$10\), plus the per-film royalty rate, which is given by: $$r = 0.002Q$$ In a competitive market, the price is equal to the cost, so the price of each tape is: $$P = 10 + r$$
02

Substitute and find the equilibrium price and quantity

Replace r in the pricing equation with its functional form and substitute it into the demand curve equation to find the equilibrium quantity: $$Q_d = 1050 - 50(10 + 0.002Q)$$ Solve for Q: $$Q = 600$$ Now compute the equilibrium price: $$P = 10 + 0.002Q = 10 + 0.002(600) = 11.2$$
03

Calculate the number of tape firms

Each firm can copy 5 tapes per day, so we can find the number of firms by dividing the total quantity by the quantity per firm: $$\text{Number of Firms} = \frac{Q}{5} = \frac{600}{5} = 120$$
04

Compute the per-film royalty rate

Finally, calculate the per-film royalty rate using the given formula: $$r = 0.002Q = 0.002(600) = 1.2$$ So, in long-run equilibrium, the price of copied tapes is \(\$11.2\), the quantity is 600, there are 120 tape firms, and the per-film royalty rate is \(\$1.2\). b. Finding the new long-run equilibrium price, quantity, number of firms, and per-film royalty rate when demand increases
05

Write the new demand equation and restate the cost equation

The new demand curve is given by: $$Q_d = 1600 - 50P$$ The cost equation is still: $$P = 10 + r$$
06

Substitute and find the new equilibrium price and quantity

Replace r in the price equation with its functional form, and substitute this into the new demand curve equation to find the new equilibrium quantity: $$Q_d = 1600 - 50(10 + 0.002Q)$$ Solve for Q: $$Q = 1000$$ Now compute the new equilibrium price: $$P = 10 + 0.002Q = 10 + 0.002(1000) = 12$$
07

Calculate the new number of tape firms

Divide the new total quantity by the quantity per firm to find the new number of firms: $$\text{Number of Firms} = \frac{Q}{5} = \frac{1000}{5} = 200$$
08

Compute the new per-film royalty rate

Calculate the new per-film royalty rate using the given formula: $$r = 0.002Q = 0.002(1000) = 2$$ So, when demand increases, the new long-run equilibrium price is \(\$12\), the quantity is 1000, there are 200 tape firms, and the per-film royalty rate is \(\$2\). c. Graphing the long-run equilibria and calculating the increase in producer surplus Since the graphical part of this problem cannot be shown here, the student should graph the demand and cost curves for both scenarios and compute the increase in producer surplus by calculating the area between the cost curves under the higher demand curve. d. Verifying that the increase in producer surplus equals the increase in royalties paid To verify this, we will now calculate the increase in royalties paid as the output expands incrementally from its level in part (a) to its level in part (b).
09

Calculate the change in royalties for each tape

First, find the difference in the per-film royalty rate between parts (a) and (b): $$\Delta r = r_{b} - r_{a} = 2 - 1.2 = 0.8$$
10

Calculate the change in royalties for 1 unit increase in quantity

Now, calculate the change in royalties paid for one additional tape: $$\Delta \text{Royalties} = \Delta r * \Delta Q = 0.8 * 1 = 0.8$$
11

Compare the increase in producer surplus to the increase in royalties paid

The increase in producer surplus should be equal to the increase in royalties paid. If our graph of producer surplus change and the incremental increase of royalties paid matches, we can conclude that the relationship holds true in this exercise.

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Most popular questions from this chapter

Suppose that the demand for broccoli is given by $$d=1,000-5 P$$ where \(Q\) is quantity per year measured in hundreds of bushels and \(P\) is price in dollars per hundred bushels. The long-run supply curve for broccoli is given by $$Q=4 \mathrm{P}-80$$ a. Show that the equilibrium quantity here is \(Q=400\). At this output, what is the equilib rium price? How much in total is spent on broccoli? What is consumer surplus at this equilibrium? What is producer surplus at this equilibrium? b. How much in total consumer and producer surplus would be lost if \(Q=300\) instead of \(Q=400 ?\) c. Show how the allocation between suppliers and demanders of the loss of total consumer and producer surplus described in part (b) depends on the price at which broccoli is sold. How would the loss be shared if \(P=140 ?\) How about if \(P=95 ?\) d. What would be the total loss of consumer and producer surplus if \(Q=450\) rather than \(Q=400 ?\) Show that the size of this total loss also is independent of the price at which the broccoli is sold.

The domestic demand for portable radios is given by $$\mathrm{g}=5,000-100 \mathrm{P}$$ where price \((P)\) is measured in dollars and quantity \((Q)\) is measured in thousands of radios per year. The domestic supply curve for radios is given by $$\mathrm{Q}=150 \mathrm{P}$$ 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 getting 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 dead weight 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.

In our analysis of tariffs we assumed that the country in question faced a perfectly elastic supply curve for imports. Now assume this country faces a positively sloped supply curve for imported goods. a. Show graphically how the level of imports will be determined. b. Use your graph from part (a) to demonstrate the effects of a tariff in this market. c. Carefully identify the sources of the various changes in consumer and producer surplus that are brought about by the tariff in part (b). d. Show how the dead weight losses brought about by the tariff in this case will depend on the elasticity of demand and the elasticities of supply of domestic and imported goods.

The handmade snuffbox industry is composed of 100 identical firms, each having short-run total costs given by $$S T C=0.5 q^{2}+l O q+5$$ and short-run marginal costs 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 sup ply 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.

Suppose that the government institutes a \(\$ 5.50\) per-film tax on the film copying industry described in Problem 15.3 a. Assuming that the demand for copied films is that given in part (a) of Problem \(15.3,\) how will this tax affect the market equilibrium? b. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? c. Show that the loss of producer surplus as a result of this tax is borne completely by the film studios. Explain your result intuitively.
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