Question

Mr. Ellis sells "Buzzbee Frisbess" door-to-door. In an average month, he sells 500 frisbees at a price of \(\$ 5\) each. Next month, his company is planning an employee contest whereby if any employee sells 1,000 frisbees, he will receive an extra two weeks vacation with pay. Never one to work too hard, Mr. Ellis decides that instead of trying to push \(\$ 5\) frisbees on unwilling customers for 12 hours a day, he will maintain his normal work schedule of 8 hours each day. His strategy is to lower the price which he charges his customers. If demand elasticity, \(\mathrm{e}=-3\), what price should Mr. Ellis charge in order to sell 1000 "Buzzbee Frisbees." Use average values for \(\mathrm{P}\) and \(\mathrm{Q}\).

Short answer

To sell 1000 "Buzzbee Frisbees" while maintaining his normal work schedule, Mr. Ellis should lower the price to approximately \$3.33 each. This price adjustment is based on a demand elasticity of -3 and his current selling rate of 500 frisbees per month at \$5 each.

Step-by-step solution

Find the percentage change in quantity

Since Mr. Ellis wants to sell 1000 frisbees, we can find the percentage change in quantity by comparing the desired quantity to the current quantity (which is 500). \[\%\Delta Q = \frac{\text{New Quantity} - \text{Current Quantity}}{\text{Current Quantity}}\times 100\] \[\%\Delta Q = \frac{1000 - 500}{500} \times 100\] \[\%\Delta Q = 100\%\]

Find the percentage change in price

We know the price elasticity of demand (E) is -3, and we calculated the percentage change in quantity to be 100%. Now we can find the percentage change in price using the formula: \[E = \frac{(\%\Delta Q)}{(\%\Delta P)}\] Let's solve for \(\%\Delta P\): \[\%\Delta P = \frac{(\%\Delta Q)}{E}\] \[\%\Delta P = \frac{100}{-3}\] \[\%\Delta P \approx -33.33\%\]

Calculate the new price

Now that we have the percentage change in price, we can calculate the new price Mr. Ellis should charge for the frisbees: \[\text{New Price} = \text{Current Price} \times (1 + \frac{\text{\%\Delta P}}{100})\] \[\text{New Price} = 5 \times (1 - 0.3333)\] \[\text{New Price} \approx \$3.33\] Thus, Mr. Ellis should charge approximately \(\$3.33\) each to sell 1000 "Buzzbee Frisbees".

Key concepts

Understanding Quantity Demanded

Quantity demanded refers to the total number of units of a specific good or service that consumers are willing and able to purchase at a given price over a specific time period. In Mr. Ellis's case, the current quantity demanded is 500 frisbees per month. Next month, he wants to increase the quantity demanded to 1,000 frisbees.
To find out how much demand needs to change, one can calculate the percentage change in quantity demanded. This involves comparing the new desired quantity with the current quantity:

• Current Quantity: 500 frisbees
• Desired Quantity: 1,000 frisbees

By using the formula: \[\%\Delta Q = \frac{\text{New Quantity} - \text{Current Quantity}}{\text{Current Quantity}}\times 100 \]Ellis learns that the demand needs to increase by 100%. This means he has to find a way to make sure his sales double in the upcoming month.

Interpreting Percentage Change

Percentage change is a useful measure to compare how much a variable has increased or decreased in relation to its original value. In the context of demand and price elasticity, it helps understand the responsiveness of quantity demanded to changes in price.
For Mr. Ellis, the goal is to increase the quantity demanded by 100% from 500 frisbees to 1,000 frisbees. This requires calculating the percentage change in price, considering the price elasticity of demand.
Using the elasticity formula: \[E = \frac{(\%\Delta Q)}{(\%\Delta P)}\]We have:

• \( E = -3 \)
• \( \%\Delta Q = 100\% \)

These equations highlight the interaction between price and demand, and demonstrate how significant this percentage measure can be for making informed pricing decisions to drive sales.

The Role of Price Change

Price change refers to the adjustment made to the selling price of goods or services to influence consumer purchase decisions, often aiming to increase demand.
In the scenario of Mr. Ellis, a price adjustment is necessary to achieve the desired increase in sales volume. Calculating the percentage change in price helps in determining the necessary adjustment in pricing strategy. Given:

• Price Elasticity of Demand, \( E = -3 \)
• Desired \( \%\Delta Q = 100\% \)

The formula \[\%\Delta P = \frac{(\%\Delta Q)}{E} = \frac{100}{-3} \approx -33.33\%\]reveals that Mr. Ellis needs to lower prices by approximately 33.33% in order to double the sales quantity. Therefore, understanding price changes is crucial for devising effective pricing strategies to meet specific sales targets.

Consumer Behavior Insights

Consumer behavior explores how individuals make decisions to spend their available resources on consumption-related items. This involves understanding how factors such as price influence the quantity demanded of a product.
For Mr. Ellis, understanding consumer behavior means knowing that lowering prices can encourage more purchases. Price elasticity of demand measures this responsiveness. A value of -3 suggests that the quantity demanded is highly responsive to price changes.
Through the lens of price elasticity:

• If demand is elastic (like Mr. Ellis's case), a small decrease in price leads to a large increase in quantity demanded.
• Understanding elasticity helps predict consumer reactions to changes in price, aiding in setting optimal prices.

By accurately gauging consumer responses, Mr. Ellis can effectively adjust his pricing strategy to boost sales and achieve business goals.