Problem 11 - Applied Calculus

Question

School organizations raise money by selling candy door to door. The table shows \(p\), the price of the candy, and \(q\), the quantity sold at that price. $$\begin{array}{c|c|c|c|c|c|c|c} \hline p & \$ 1.00 & \$ 1.25 & \$ 1.50 & \$ 1.75 & \$ 2.00 & \$ 2.25 & \$ 2.50 \\\ \hline q & 2765 & 2440 & 1980 & 1660 & 1175 & 800 & 430 \\ \hline \end{array}$$ (a) Estimate the elasticity of demand at a price of $$\$ 1.00$$. At this price, is the demand elastic or inelastic? (b) Estimate the elasticity at each of the prices shown. What do you notice? Give an explanation for why this might be so. (c) At approximately what price is elasticity equal to 1 ? (d) Find the total revenue at each of the prices shown. Confirm that the total revenue appears to be maximized at approximately the price where \(E=1\).

Step-by-Step Solution

Answer

Demand is inelastic at \$1.00. Elasticity is near 1 at \$1.50. Revenue maximizes at \$1.25.

Step 1: Understanding Elasticity of Demand

Elasticity of demand measures how the quantity demanded of a good changes in response to changes in price. It is computed as: \( E = \frac{\Delta q}{\Delta p} \times \frac{p}{q} \), where \( \Delta q \) is the change in quantity and \( \Delta p \) is the change in price. When \( E > 1 \), demand is elastic; when \( E < 1 \), it is inelastic; and when \( E = 1 \), it is unitary elastic.

Step 2: Calculate Elasticity at \( p = \$1.00 \)

To estimate elasticity at \( p = \\(1.00 \), use the change from \( p = \\)1.00 \) to \( p = \\(1.25 \). The initial quantity \( q_1 = 2765 \) and \( q_2 = 2440 \) when price changes from \\)1.00 to \\(1.25. Thus, \( \Delta q = 2440 - 2765 = -325 \). \( \Delta p = 1.25 - 1.00 = 0.25 \). Now calculate \( E = \frac{-325}{0.25} \times \frac{1.00}{2765} = -0.47 \). Thus, the demand is inelastic at \\)1.00.

Step 3: Calculate Elasticity for Each Price

Similarly, compute elasticity for each subsequent pair of prices: \( \\(1.25 \) to \( \\)1.50\), etc., using the formula \( E = \frac{\Delta q}{\Delta p} \times \frac{p}{q} \). Calculate for each interval:- \\(1.25: E \approx -0.83- \\)1.50: E \approx -1.03- \\(1.75: E \approx -1.35- \\)2.00: E \approx -1.96- \\(2.25: E \approx -2.81- \\)2.50: E \approx -5.12.

Step 4: Locate Price where Elasticity is 1

The demand elasticity is closest to 1 at \( p = \$1.50 \) where \( E \approx -1.03 \). This indicates approximate unitary elasticity.

Step 5: Calculate Total Revenue at Each Price

Total revenue \( R \) is calculated as \( R = p \times q \). Compute for each price:- \\(1.00: \( R = 1.00 \times 2765 = \\)2765 \)- \\(1.25: \( R = 1.25 \times 2440 = \\)3050 \)- \\(1.50: \( R = 1.50 \times 1980 = \\)2970 \)- \\(1.75: \( R = 1.75 \times 1660 = \\)2905 \)- \\(2.00: \( R = 2.00 \times 1175 = \\)2350 \)- \\(2.25: \( R = 2.25 \times 800 = \\)1800 \)- \\(2.50: \( R = 2.50 \times 430 = \\)1075 \).The revenue is maximized at \( p = \$1.25 \), slightly before the price with unitary elasticity.

Key Concepts

Total Revenue

Total revenue (TR) is the total amount of money that a firm receives from selling its goods or services, calculated by multiplying the price per unit (p) by the quantity sold (q). For the candy sale example, total revenue helps to determine which price level generates the most income for the school organization.

When analyzing the total revenue across different prices, one can observe how different levels of pricing affect the earnings:

• At $1.00, TR is 2765 since they sold 2765 candies.
• When the price increases to $1.25, the revenue increases to 3050, showing improved earnings despite selling fewer candies.
• The revenue begins to decrease with further price increases, as shown at $1.50 with TR of 2970.

Thus, a key insight is that the total revenue doesn't always rise with price increases, as fewer sales can result at higher price points. The management of pricing is crucial for maximizing revenue, and as seen in the data, it peaks at $1.25 before demand becomes significantly more price-sensitive.

Inelastic Demand

Inelastic demand occurs when a change in price leads to a minor change in the quantity demanded. This means that consumers are relatively insensitive to price changes, continuing to buy almost the same quantity even if price increases. The calculated elasticity at $1.00 was -0.47, which is less than 1 in absolute value, indicating inelastic demand.

When demand is inelastic, the total revenue tends to move in the same direction as the price. For example, when the price increased from $1.00 to $1.25, despite the decrease in quantity from 2765 to 2440, the total revenue still increased from 2765 to 3050. This showcases the classic case of inelastic demand where businesses can gain by increasing prices without a significant drop in sales volume.

Understanding inelastic demand can help businesses strategize prices effectively when attempting to maximize revenues without losing much in terms of quantity sold.

Unitary Elasticity

Unitary elasticity occurs when a percentage change in price leads to an equal percentage change in quantity demanded. This results in total revenue remaining unchanged when prices change. An elasticity value of exactly 1 indicates unitary elasticity.

In our example, elasticity is closest to -1 at the price of $1.50, calculated to be -1.03. Here, the quantity sold changes proportionately with price changes, meaning revenue generation doesn't fluctuate much around this point. Unitary elasticity is critical for pricing strategies, as it indicates a balance point where any increase or decrease in price doesn't affect the overall revenue significantly.

• At $1.50, the revenue was 2970, slightly less than the revenue at $1.25.
• It showcases that companies need to decide what kind of demand elasticity they are targeting to optimize profit.

Recognizing this balance can be essential for making informed decisions about setting or adjusting prices in line with desired revenue outcomes.