Question

The director of a theater company in a small college town is considering changing the way he prices tickets. He has hired an economic consulting firm to estimate the demand for tickets. The firm has classified people who go to the theater into two groups and has come up with two demand functions. The demand curves for the general public \(\left(Q_{g p}\right)\) and students \(\left(Q_{s}\right)\) are given below: \\[ \begin{aligned} Q_{8 p} &=500-5 P \\ Q_{s} &=200-4 p \end{aligned} \\] a. Graph the two demand curves on one graph, with \(P\) on the vertical axis and \(Q\) on the horizontal axis. If the current price of tickets is \(\$ 35,\) identify the quantity demanded by each group. b. Find the price elasticity of demand for each group at the current price and quantity. c. Is the director maximizing the revenue he collects from ticket sales by charging \(\$ 35\) for each ticket? Explain. d. What price should he charge each group if he wants to maximize revenue collected from ticket sales?

Short answer

a. For students, quantity demanded at $35 is 60 tickets and for general public is 325 tickets. b. The price elasticity for students and general public are both less than 1, indicating that demand is inelastic. c. Whether $35 maximizes revenue depends on the elasticity of demand. Since price elasticity of demand for both groups is inelastic, a price increase will increase total revenue, hence $35 may not the revenue maximizing price. d. To maximise revenue, find the maximum of the total revenue curve obtained from plotting ticket price against total revenue for each group.

Step-by-step solution

Graphing the demand curves and identifying the current quantity demanded

The two demand functions given are linear equations, so they can be graphed as straight lines. Plugin price P = 35 into both functions to find the quantity demanded by each group.

Finding the price elasticity of demand

The price elasticity of demand can be calculated using the formula: \(\epsilon = \frac{dQ}{dP} \times \frac{P}{Q}\). Since the quantity demanded is a function of price, we can use the formulas \(Q_{s} = 200 - 4P\) and \(Q_{g p} = 500 - 5P\) to find the derivative \(\frac{dQ}{dP}\). We then substitute the current price and corresponding quantity into the formula to find the price elasticity of demand for each group.

Check if the director is maximizing his revenue

Revenue is product of price and quantity. To find if the director's choice of $35 is maximizing his revenue, let's calculate the revenue at this price: \(R = P \times Q = 35 \times (Q_{s} + Q_{g p})\). Now, to maximize revenue if P != 0 and elasticity is bigger than 1, a price decrease will increase revenue. If elasticity is less than 1, a price increase will raise revenue.

Finding the revenue maximizing price using total revenue test

The total revenue test is a method of determining whether demand is elastic or inelastic. If an increase in price brings about an increase in total revenue, demand is classified as inelastic. If a decrease in price increases total revenue, demand is elastic. Plot the total revenue curve using the demand functions to find the prices that maximize total revenue.

Key concepts

Price Elasticity of Demand

Price elasticity of demand measures how the quantity demanded of a good responds to a change in its price. This concept can be understood using the formula: \[\epsilon = \frac{dQ}{dP} \times \frac{P}{Q}\] Here, \(dQ/dP\) is the derivative of the demand function with respect to price, \(P\) is the price, and \(Q\) is the quantity demanded. The result tells us how sensitive consumers are to price changes. - **Elastic Demand**: If the elasticity is greater than 1, demand is elastic, meaning consumers are highly responsive to price changes. A slight decrease in price can significantly increase quantity demanded, and revenue will likely increase. - **Inelastic Demand**: If the elasticity is less than 1, demand is inelastic. Consumers are less responsive to price changes. A price increase will increase total revenue since the decrease in quantity demanded won't offset the higher price. - **Unit Elastic**: If elasticity equals 1, changes in price do not affect total revenue. In our example, to calculate elasticity at the current price of $35 for each group, one would substitute the respective derivatives and values into the formula. This helps the theater director understand how sensitive each group is to changes in ticket prices.

Revenue Maximization

Revenue maximization means selling at a price where total revenue peaks. Total revenue (TR) is calculated as price \(P\) multiplied by quantity \(Q\), or \[TR = P \times Q\]. To find the revenue-maximizing price, it’s crucial to determine the demand's elasticity at different prices. The relationship between price elasticity and revenue is pivotal: - If demand is elastic (elasticity > 1), lowering the price can increase total revenue because the percentage drop in price will lead to a larger percentage increase in quantity demanded. - Conversely, if demand is inelastic (elasticity < 1), increasing the price maximizes revenue, as the drop in quantity demanded is proportionally smaller than the price increase. The theater director can use these insights to consider whether the current ticket price is optimal or if adjusting it could help achieve maximum revenue from each group.

Demand Function

A demand function shows the relationship between the quantity demanded of a good and its price. In mathematical terms, it is expressed as \(Q = a - bP\), where \(Q\) is the quantity demanded, \(P\) is the price, \(a\) is the intercept, and \(b\) represents the rate of change. In the theater's case, we have two distinct demand functions: - General Public: \[Q_{gp} = 500 - 5P\] - Students: \[Q_{s} = 200 - 4P\] Each function indicates how quantity demanded changes with price adjustments for different groups. Graphing these linear functions with price on the vertical axis and quantity on the horizontal axis shows where the demand curves touch, indicating the market situation for the sets of customers. Understanding and using the demand function helps the director anticipate how different prices will affect attendance and overall ticket sales.

Price Discrimination

Price discrimination occurs when a seller charges different prices to different consumer groups for the same good or service, based on their willingness to pay. This strategy is profitable if certain conditions are met: 1. **Market Control**: The seller must have some control over the price and product. 2. **Segmented Markets**: Markets must be distinguishable and separated so that consumers cannot simply resell the product to one another. 3. **Varying Price Sensitivity**: Different groups must exhibit different price elasticities of demand. For the theater director, applying price discrimination might mean charging different prices to students and the general public based on their respective demand elasticities. Leveraging the calculated elasticities, the director can potentially increase revenue by maximizing prices according to each group's willingness to pay. Moreover, price discrimination allows the theater to tap into the maximum value that different consumer segments place on the tickets, boosting overall profitability.