Problem 43 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

A cruise line estimates that when each deluxe balcony stateroom on a particular cruise is priced at \(p\) thousand dollars, then \(q\) tickets for staterooms will be demanded by travelers, where \(q=300-0.7 p^{2}\). a. Find the elasticity of demand for the stateroom tickets. b. When the price is \(\$ 8,000(p=8)\) per stateroom, should the cruise line raise or lower the price to increase total revenue?

Step-by-Step Solution

Answer

Elasticity is approximately -0.351. Since demand is inelastic, the cruise line should raise the price to increase total revenue.

Step 1: Write the demand function

The demand function is given by the equation: \[ q = 300 - 0.7p^2 \]

Step 2: Find the derivative of the demand function with respect to price

To find the elasticity of demand, we need the derivative of the demand function with respect to price, denoted as \( \frac{dq}{dp} \). The demand function is \[ q = 300 - 0.7p^2 \]. Differentiating with respect to \( p \), we get: \[ \frac{dq}{dp} = -1.4p \]

Step 3: Use the elasticity formula

Elasticity of demand \( E \) is given by the formula: \[ E = \left(\frac{p}{q}\right) \left(\frac{dq}{dp}\right) \]. Substituting \( q = 300 - 0.7p^2 \) and \( \frac{dq}{dp} = -1.4p \), we get: \[ E = \left(\frac{p}{300 - 0.7p^2}\right) \left(-1.4p\right) \]

Step 4: Simplify the elasticity formula

Simplify the expression for elasticity: \[ E = \frac{-1.4p^2}{300 - 0.7p^2} \]

Step 5: Evaluate the elasticity at \( p = 8 \)

Substitute \( p = 8 \) into the elasticity expression: \[ E = \frac{-1.4(8)^2}{300 - 0.7(8)^2} \] Calculate the values: \[ E = \frac{-1.4(64)}{300 - 0.7(64)} \] Simplify further: \[ E = \frac{-89.6}{300 - 44.8} \] \[ E = \frac{-89.6}{255.2} \] \[ E \approx -0.351 \]

Step 6: Interpret the elasticity value

Since the elasticity \( E \approx -0.351 \), it is less than 1 in absolute value. This means the demand is inelastic at the price \( p = 8 \).

Step 7: Determine pricing strategy

When demand is inelastic (\( |E| < 1 \)), increasing the price will increase total revenue. Therefore, the cruise line should raise the price to increase total revenue.

Key Concepts

demand function

The demand function describes the relationship between the price of a product and the quantity demanded by consumers. In this problem, the demand function is given by: \[ q = 300 - 0.7p^2 \]Here, \(q\) represents the number of deluxe balcony stateroom tickets demanded, and \(p\) is the price of each stateroom in thousand dollars. This quadratic function indicates that as the price increases, the quantity demanded decreases. Understanding the demand function is crucial because it forms the foundation for analyzing how changing prices affect demand.

derivative of demand function

To find how responsive the quantity demanded is to price changes, we need the derivative of the demand function. The derivative provides the rate at which the demand changes with respect to the price. For our given demand function \( q = 300 - 0.7p^2 \), we differentiate with respect to\( p \): \[ \frac{dq}{dp} = -1.4p \].This derivative, \(-1.4p\), shows that for each unit increase in price, the demand decreases by \(1.4p\) units. This rate is essential for calculating the price elasticity of demand, helping us determine the sensitivity of demand to price changes.

price elasticity

Price elasticity of demand measures how much the quantity demanded of a good responds to changes in its price. It is calculated using the formula: \[ E = \left(\frac{p}{q}\right) \left(\frac{dq}{dp}\right) \].For our demand function, substituting \( q = 300 - 0.7p^2 \) and \( \frac{dq}{dp} = -1.4p \), we get:\[ E = \left(\frac{p}{300 - 0.7p^2}\right) \left(-1.4p\right) \].Simplifying,\[ E = \frac{-1.4p^2}{300 - 0.7p^2} \].At \( p = 8 \), we substitute to find the elasticity: \[ E = \frac{-89.6}{255.2} \approx -0.351 \].Because \(|E| < 1\), the demand is considered inelastic, meaning quantity demanded isn't very responsive to price changes.

pricing strategy

Based on the elasticity of demand, businesses can formulate their pricing strategy. If demand is inelastic (\(|E| < 1\)), increasing the price will lead to an increase in total revenue since the percentage decrease in demand is less than the percentage increase in price. For our cruise line, since \( |E| \approx 0.351 \), the demand is inelastic at a price of \$8,000. Therefore, the cruise line should increase the price to boost total revenue. Understanding elasticity helps businesses make informed decisions about pricing to optimize their revenues.