Question

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 perfilm royalty rate \((r)\) is an increasing function of total industry output \((Q):\) \\[r=0.002 Q.\\] \\[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

In long-run equilibrium: a) Price: $11, Quantity: 500, Firms: 100, Royalty: $1; b) Price: $12, Quantity: 1000, Firms: 200, Royalty: $2.

Step-by-step solution

Understanding Long-run Equilibrium

In a perfectly competitive market, the price equals the average cost in long-run equilibrium. The average cost includes both the production cost and royalty. The average cost per tape is $10, and we must add the royalty to find the equilibrium price: \(P = 10 + r.\)

Finding Equilibrium Using Demand and Supply

The demand curve is \(Q = 1050 - 50P.\) In equilibrium, the price equals the average cost: \(P = 10 + 0.002Q.\) Substitute \(P\) in the demand equation: \(1050 - 50(10 + 0.002Q) = Q.\) Solve for \(Q\) to get the equilibrium quantity.

Solving Equilibrium Equation

Rearrange the equilibrium equation: \(1050 - 500 - 0.1Q = Q.\) Simplify to find \(1050 - 500 = 1.1Q,\) \(Q = \frac{550}{1.1} \approx 500.\) So, the equilibrium quantity is 500 tapes.

Finding Equilibrium Price and Number of Firms

Substitute \(Q = 500\) back into \(P = 10 + 0.002Q\) to find \(P = 11.\) Each firm produces 5 tapes, so there are \(\frac{500}{5} = 100\) firms. The royalty rate is \(r = 0.002 \times 500 = 1.\)

New Long-run Equilibrium for Increased Demand

For part (b), use the new demand curve \(Q = 1600 - 50P.\) With \(P = 10 + 0.002Q,\) substitute \(P\): \(1600 - 50(10 + 0.002Q) = Q.\) Solve for \(Q\) to find the new equilibrium quantity.

Calculate Equilibrium for Increased Demand

Rearrange to \(1600 - 500 - 0.1Q = Q.\) \(1600 - 500 = 1.1Q,\) giving \(Q = \frac{1100}{1.1} \approx 1000.\) So, the new equilibrium quantity is 1000 tapes.

Find New Price and Number of Firms

Substitute \(Q = 1000\) back into \(P = 10 + 0.002Q\) to find \(P = 12.\) With 5 tapes per firm, there are \(\frac{1000}{5} = 200\) firms. The royalty rate is \(r = 0.002 \times 1000 = 2.\)

Key concepts

Long-run Equilibrium

In the context of perfect competition, long-run equilibrium is an important concept where firms adjust to a point where economic profit is zero. This means that the market price equals the minimum average cost. Companies in a perfectly competitive market are "price takers," which means they don't have the power to set prices and must accept the market rate. In the long-run, any economic profit attracts new firms, increasing supply and lowering prices until profits are wiped out. Conversely, if firms incur losses, some leave the market, decreasing supply and raising prices until remaining firms cover their average costs.

In the exercise, you found this equilibrium by setting the price equal to the average cost, which included both production and royalty costs for tape copying. For instance, with a production cost of $10 per tape, the equilibrium price was determined through the equation \( P = 10 + r \), where \( r \) is the royalty that depends on total output \( Q \). In both scenarios provided in the exercise, the number of firms adjusts to ensure that each can just cover its costs.

Demand Curve

The demand curve in economics represents the relationship between the price of a good and the quantity demanded. In the exercise, the demand curves are given as linear equations, for example, \( Q = 1,050 - 50P \) for the initial situation and \( Q = 1,600 - 50P \) when demand increases. These equations show that as the price \( P \) changes, the quantity \( Q \) demanded by consumers changes inversely.

Understanding the demand curve helps us predict how different factors—like changes in consumer preferences or income—can affect quantity demanded. For perfectly competitive markets like the tape copying industry, the demand curve is crucial for determining equilibrium. By substituting equilibrium price expressions into the demand equation, you calculate the equilibrium quantities directly. This process lets you see how shifts in demand influence equilibrium price and quantity.

Producer Surplus

Producer surplus is the difference between the revenue producers receive and the minimum amount they would accept for supplying a given quantity. It is a measure of producers' wellbeing and market efficiency. In the exercise, the producer surplus changes as market conditions shift, indicating changes in overall industry profit.

More precisely, when demand increases, the producer surplus grows, as demonstrated during the shift from part (a) to part (b) of the exercise. Calculating this increase involves understanding the change in equilibrium points. Producers gain because they sell more at higher prices, reflecting better market conditions. However, part of this benefit is captured by the studios through increased royalties, which rise as higher output leads to higher costs under \( r = 0.002Q \).

Supply and Demand Analysis

Supply and demand analysis is a primary tool used in economics to determine how markets function. It involves understanding how various factors influence the supply and demand of goods, and thus affect the equilibrium price and quantity in a market. This analysis is crucial in perfect competition, as seen in the videotape copying industry.

In perfect competition, supply and demand curves determine the market equilibrium. Producers adjust output to meet market price, while consumers adjust demand based on their willingness to pay. The demand curve stories from the exercise show shifts due to increased appetite for tapes, resulting in different prices and quantities at new equilibrium points.

Moreover, supply and demand analysis helps in visualizing changes in producer surplus and the impact of market adjustments. When analyzing shifts, such as those caused by the introduction of a tax, you can explore consumer and producer burden by examining how these changes affect equilibrium and welfare in the market.