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

(a) Inelastic at $1.00 with E ≈ 0.93. (b) Elasticity increases with price; becomes more elastic. (c) Elasticity equals 1 around $1.50. (d) Revenue maximizes near where E=1.

Step 1: Understand Elasticity of Demand

Elasticity of demand measures how much the quantity demanded responds to changes in price. It is calculated as \[ E = \left( \frac{\% \text{ change in quantity}}{\% \text{ change in price}} \right) \]If \(E > 1\), demand is elastic. If \(E < 1\), demand is inelastic. If \(E = 1\), demand has unit elasticity.

Step 2: Compute Elasticity for Price $1.00

The elasticity at any price \(p_i\) is estimated using the midpoint formula:\[E = \frac{(q_2 - q_1) / ((q_1 + q_2)/2)}{(p_2 - p_1) / ((p_1 + p_2)/2)}\]For \(p = \\(1.00\) (\(q = 2765\)) and \(p = \\)1.25\) (\(q = 2440\)), we have:\[E = \frac{(2440 - 2765) / ((2765 + 2440) / 2)}{(1.25 - 1.00) / ((1.25 + 1.00) / 2)} = \frac{-325 / 2602.5}{0.25 / 1.125} \approx -0.93\]Since \(E = 0.93 < 1\), the demand is inelastic.

Step 3: Estimate Elasticity for Other Prices

Following a similar process as Step 2, calculate the elasticity for each subsequent price:- \(E_{1.25} = \frac{1980 - 2440}{(2440 + 1980)/2} / \frac{1.50 - 1.25}{(1.50 + 1.25)/2} \approx -1.23\)- \(E_{1.50} = \frac{1660 - 1980}{(1980 + 1660)/2} / \frac{1.75 - 1.50}{(1.75 + 1.50)/2} \approx -1.47\)- Continue similarly up to \(E_{2.25}\).Notice that elasticity increases with price, shifting from inelastic to elastic.

Step 4: Analyze Elasticity at Each Price

As price increases, demand becomes more elastic. A possible explanation is that at higher prices, consumers are more responsive to changes since the relative cost change is more significant. Lower prices result in less sensitivity because the incremental cost change is smaller relative to the overall price.

Step 5: Find Price where Elasticity Equals 1

By examining the calculated elasticities, identify the price range where elasticity transitions to 1. This crossover point approximates where demand elasticity becomes unit elastic.

Step 6: Calculate Total Revenue for Each Price

Total revenue \(R\) at each price is calculated as \(R = p \times q\).- For \(p = \$1.00\), \(R = 1.00 \times 2765 = 2765\).- Continue this calculation for each price point presented.Compare these totals and observe which price range maximizes revenue.

Step 7: Confirm Maximum Revenue at Elasticity Equal to 1

Given that total revenue appears maximized where elasticity equals 1, verify this relationship by comparing calculated revenues with their corresponding elasticities. The price where elasticity is closest to 1 should have the highest revenue, confirming the relationship.

Key Concepts

Price and Quantity Relationship

In the world of economics, understanding the interplay between price and quantity is vital. When we alter the price of a product, it inevitably affects the quantity of that product that consumers are willing to purchase. This relationship is often depicted in demand curves, where typically, a lower price leads to higher quantity demands, and vice versa.

In our candy-selling scenario, as prices increase from $1.00 to $2.50, the quantity sold decreases incrementally. For example, when the price is low at $1.00, the highest quantity of candy, 2765, is sold. However, as the price rises to $2.50, only 430 candies are sold. This inverse relationship between price and quantity is a clear representation of the law of demand:

• Higher prices mean fewer consumers will buy the product.
• Lower prices generally increase consumer purchase.

Keeping this relationship in mind helps businesses and economists predict consumer behavior. Adjusting price points strategically allows maximizing sales volume or revenue under certain economic conditions.

Inelastic and Elastic Demand

Understanding the concept of inelastic and elastic demand is crucial in economics. Elasticity of demand indicates how sensitive quantity demanded is to a change in price. If demand is elastic, a small change in price leads to a large change in the quantity demanded. Conversely, inelastic demand means consumers are less responsive to price changes.

In our exercise, at a price of $1.00, the elasticity was calculated as approximately -0.93, which is less than 1. This indicates inelastic demand, meaning consumers do not significantly change their purchase quantities with price changes.

• Inelastic Demand ( |E| < 1 ): Consumers are not sensitive to price changes.
• Elastic Demand ( |E| > 1 ): Consumers are sensitive to price changes.
• Unit Elasticity ( |E| = 1 ): Proportional response to price changes.

As price increases along the table, the demand becomes more elastic, suggesting consumers become more price-sensitive at higher price points. This switch occurs because a higher price makes the relative cost difference more noticeable to a buyer.

Total Revenue Calculation

Total revenue is a key metric for businesses to evaluate their financial performance, especially when considering price changes. It is calculated as the product of the price per unit and the quantity sold \[ R = p \times q \].

In the exercise, you see how total revenue changes with different prices for the candy:

• For a price of \(1.00, total revenue is 2765.
• As the price increases to \)2.00, total revenue also increases, but then it starts declining from this point as prices get higher.

This showcases an important insight: total revenue first increases as prices rise, but after reaching a particular point (where elasticity is close to 1), it starts decreasing. The idea is to maximize revenue where elasticity is unitary (|E|=1), avoiding the extremes where significant drops in quantity sold can adversely affect revenue.

Midpoint Formula

To determine elasticity, we often use the midpoint formula. It provides a more precise measure compared to simple percentage changes because it takes into account the average percent changes in both quantity and price.

The formula is given by:

\[ E = \frac{(q_2 - q_1) / ((q_1 + q_2)/2)}{(p_2 - p_1) / ((p_1 + p_2)/2)} \]
• \(q_1\) and \(q_2\) are initial and final quantities.
• \(p_1\) and \(p_2\) are initial and final prices.

This formula is especially useful in avoiding the discrepancies that arise when using standard percentage change calculations. Instead of basing the percentage change off the initial or final values alone, the midpoint formula uses the average values, giving a more balanced elasticity estimate. This nuanced calculation is essential for better understanding consumer reactions to price changes over a range of price points.