Calculating the Point Elasticity of Demand for Sally’s Sea Shells:

To calculate the point elasticity of demand for Sally’s sea shells, we can use the formula for point price elasticity of demand, which is:

�=���×���E=qdq​×dpp​

Where:

• $E$ represents the point price elasticity of demand.
• $dq$ represents the change in quantity demanded.
• $q$ represents the initial quantity demanded.
• $p$ represents the initial price.
• $dp$ represents the change in price.

In this case, Sally’s average daily price is $8.75, and she sells 71 quantities at that price. We have the following values:

$q=71$ (initial quantity) $p=8.75$ (initial price)

Now, we need to find the change in quantity demanded and the change in price to calculate the point elasticity.

Let’s assume Sally changes her price by a small amount, say $dp=-0.01$ (a decrease of $0.01). To find the corresponding change in quantity demanded ($dq$), we can use the slope of her regression line, which is 62:

$dq=\text{Slope}\times dp=62\times (-0.01)=-0.62$

Now, we can plug these values into the elasticity formula:

�=−0.6271×8.75−0.01E=71−0.62​×−0.018.75​

Simplifying the equation:

�=0.6271×8.750.01=0.6271×875=43.7571E=710.62​×0.018.75​=710.62​×875=7143.75​

Now, calculating this value:

$E\approx 0.6162$

So, the point price elasticity of demand for Sally’s sea shells is approximately 0.6162.

2. Determining the Required Price Elasticity for Optimal Pricing:

To determine the required price elasticity of demand for optimal pricing, we need to consider the seller’s current unit margin rate, which is 20 percent. The relationship between price elasticity ($E$), unit margin rate ($UMR$), and markup rate ($MR$) is given by the following formula:

�=1���×��E=UMR1​×MR

Where:

• $E$ represents the price elasticity of demand.
• $UMR$ represents the unit margin rate.
• $MR$ represents the markup rate.

In this case, the seller’s unit margin rate ($UMR$) is 20 percent or 0.20.

Now, let’s solve for $MR$:

�=10.20×��E=0.201​×MR

Simplifying the equation:

$MR=0.20\times E$

To determine the required price elasticity ($E$) for optimal pricing, we can set $MR$ equal to 1, as optimal pricing typically means that the markup rate ($MR$) is equal to 100 percent, resulting in a 100 percent profit margin. So, we have:

$1=0.20\times E$

Now, solving for $E$:

�=10.20=5E=0.201​=5

Therefore, for the seller to achieve optimal pricing with a unit margin rate of 20 percent, the price elasticity of demand ($E$) must equal 5.

In conclusion, Sally’s point price elasticity of demand for her sea shells is approximately 0.6162, and for the seller to achieve optimal pricing with a 20 percent unit margin rate, the price elasticity of demand must equal 5. Understanding elasticity is crucial for businesses like Sally’s to make informed pricing decisions and maximize their profits.