Problem 2 - Applied Calculus

Question

The elasticity of a good is \(E=2 .\) What is the effect on the quantity demanded of: (a) A \(3 \%\) price increase? (b) A \(3 \%\) price decrease?

Step-by-Step Solution

Answer

A 3% price increase decreases quantity demanded by 6%, and a 3% price decrease increases it by 6%.

Step 1: Understanding Elasticity

Elasticity measures how much the quantity demanded of a good responds to changes in price. When given elasticity, it tells us the percentage change in quantity demanded resulting from a 1% change in price.

Step 2: Applying the Formula

The formula for elasticity is given by \[ E = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}. \] Given that the elasticity \( E = 2 \), we can rearrange this formula to find \(\text{Percentage Change in Quantity Demanded} = E \times \text{Percentage Change in Price}.\)

Step 3: Calculating Effect of a Price Increase

For a \(3\%\) price increase, we substitute the elasticity and price change into the formula: \[ \text{Percentage Change in Quantity Demanded} = 2 \times 3\% = 6\%.\] This means the quantity demanded decreases by \(6\%\) because the price increased.

Step 4: Calculating Effect of a Price Decrease

For a \(3\%\) price decrease, we again use the formula: \[ \text{Percentage Change in Quantity Demanded} = 2 \times (-3\%) = -6\%.\] Since the percentage change is negative, the quantity demanded increases by \(6\%\) when the price decreases.

Key Concepts

Price Elasticity of Demand

Price elasticity of demand is a measure in economics that helps us understand how sensitive the quantity demanded of a good is to changes in its price. Essentially, it answers the question: "If the price changes, by what percentage will the quantity demanded change?" This is crucial in determining how much to adjust prices without losing customers or revenue. The concept is used widely across different industries to set optimal pricing strategies.

To express this, we use the formula for price elasticity of demand:

• The elasticity coefficient, noted as \( E \), is the ratio of the percentage change in quantity demanded to the percentage change in price.

A higher elasticity means that the quantity demanded is very responsive to price changes, while a lower elasticity suggests that demand is relatively insensitive to price changes. Understanding this can greatly impact business decisions on pricing and marketing.

Percentage Change Calculation

Calculating the percentage change is fundamental in understanding elasticity. It involves finding how much a variable has increased or decreased relative to its original value. For elasticity in economics, it helps determine how small changes in price affect demand.

The formula for percentage change is:

• \( \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\% \)

In the context of elasticity, once we know the price change as a percentage, we apply it to the elasticity formula to find how the quantity demanded will change. For example, a 3% increase in price applied to an elasticity of 2 will result in a 6% decrease in the quantity demanded. Understanding how to calculate these changes helps businesses predict consumer behavior more accurately.

Effect on Quantity Demanded

The effect on quantity demanded due to price changes is critical for businesses looking to optimize sales and revenue. In general, if a product has a high price elasticity of demand, customers are likely to stop buying it as much if the price goes up. Conversely, they are likely to buy more if the price decreases.

When prices increase:

• With an elasticity of 2, a 3% price hike means the quantity demanded will decrease by 6%.

This negative relationship shows that higher prices deter purchases as customers look for substitutes or rethink their expenses.

When prices decrease:

• The same elasticity denotes that a 3% price drop results in a 6% increase in quantity demanded.

This indicates that lower prices attract more buyers, boosting sales. Recognizing these effects allows companies to make informed decisions about pricing, helping to balance supply and demand effectively.